:: LPSPACE2 semantic presentation

begin

theorem :: LPSPACE2:1
for m, n being ( ( positive real ) ( non empty ext-real positive non negative V38() real ) number ) st (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / m : ( ( positive real ) ( non empty ext-real positive non negative V38() real ) number ) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) + (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / n : ( ( positive real ) ( non empty ext-real positive non negative V38() real ) number ) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
m : ( ( positive real ) ( non empty ext-real positive non negative V38() real ) number ) > 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE2:2
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like V164() ) PartFunc of ,) st A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like V164() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like V164() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like V164() ) PartFunc of ,) is nonnegative holds
( Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like V164() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) in REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) iff f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like V164() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

definition
let r be ( ( real ) ( ext-real V38() real ) number ) ;
attr r is geq_than_1 means :: LPSPACE2:def 1
1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( ) NORMSTR ) ;
end;

registration
cluster real geq_than_1 -> positive real for ( ( ) ( ) set ) ;
end;

registration
cluster ext-real V38() real geq_than_1 for ( ( ) ( ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;
end;

theorem :: LPSPACE2:3
for a, b, p being ( ( ) ( ext-real V38() real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) < p : ( ( ) ( ext-real V38() real ) Real) & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) <= a : ( ( ) ( ext-real V38() real ) Real) & a : ( ( ) ( ext-real V38() real ) Real) < b : ( ( ) ( ext-real V38() real ) Real) holds
a : ( ( ) ( ext-real V38() real ) Real) to_power p : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) < b : ( ( ) ( ext-real V38() real ) Real) to_power p : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;

theorem :: LPSPACE2:4
for a, b being ( ( ) ( ext-real V38() real ) Real) st a : ( ( ) ( ext-real V38() real ) Real) >= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
a : ( ( ) ( ext-real V38() real ) Real) to_power b : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) >= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE2:5
for a, b, c being ( ( ) ( ext-real V38() real ) Real) st a : ( ( ) ( ext-real V38() real ) Real) >= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ext-real V38() real ) Real) >= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
(a : ( ( ) ( ext-real V38() real ) Real) * b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power c : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = (a : ( ( ) ( ext-real V38() real ) Real) to_power c : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) * (b : ( ( ) ( ext-real V38() real ) Real) to_power c : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;

theorem :: LPSPACE2:6
for X being ( ( non empty ) ( non empty ) set )
for a, b being ( ( ) ( ext-real V38() real ) Real)
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is nonnegative & a : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
(f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) to_power a : ( ( ) ( ext-real V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power b : ( ( ) ( ext-real V38() real ) Real) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) to_power (a : ( ( ) ( ext-real V38() real ) Real) * b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:7
for X being ( ( non empty ) ( non empty ) set )
for a, b being ( ( ) ( ext-real V38() real ) Real)
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is nonnegative & a : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
(f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) to_power a : ( ( ) ( ext-real V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) (#) (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) to_power b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) to_power (a : ( ( ) ( ext-real V38() real ) Real) + b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:8
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) holds f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) to_power 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ;

theorem :: LPSPACE2:9
for seq1, seq2 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = (seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) & seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) >= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ) holds
( seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) is convergent iff seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) is convergent ) ;

theorem :: LPSPACE2:10
for seq being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence)
for n, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( abs (((Partial_Sums seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) - ((Partial_Sums seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) <= ((Partial_Sums (abs seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) - ((Partial_Sums (abs seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) & abs (((Partial_Sums seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) - ((Partial_Sums seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) <= (Partial_Sums (abs seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ;

theorem :: LPSPACE2:11
for seq, seq2 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) is convergent & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = |.((lim seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) - (seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) .| : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) holds
( seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) is convergent & lim seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Real_Sequence) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

begin

theorem :: LPSPACE2:12
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for X being ( ( non empty ) ( non empty ) set ) holds (X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:13
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for D being ( ( ) ( ) set ) holds abs (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) | D : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = (abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) | D : ( ( ) ( ) set ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ;
cluster |.f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) .| : ( ( V1() Function-like V165() ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) Function-like V163() V164() V165() ) set ) -> V1() Function-like V165() nonnegative ;
end;

theorem :: LPSPACE2:14
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is nonnegative holds
abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ;

theorem :: LPSPACE2:15
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st X : ( ( non empty ) ( non empty ) set ) = dom f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( 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 dom f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) = f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) holds
( f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func Lp_Functions (M,k) -> ( ( non empty ) ( non empty ) Subset of ) equals :: LPSPACE2:def 2
 { f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) where f is ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ex Ef being ( ( ) ( ) Element of S : ( ( ) ( ) set ) ) st
( M : ( ( Function-like V41([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) ) ) ( V1() V4([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V5(X : ( ( ) ( ) set ) ) Function-like V41([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) ) ) Element of bool [:[:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . (Ef : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & dom f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = Ef : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on Ef : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) & (abs f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V5(X : ( ( ) ( ) set ) ) Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V41([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) ) ) ( V1() V4([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V5(X : ( ( ) ( ) set ) ) Function-like V41([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) ) ) Element of bool [:[:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

theorem :: LPSPACE2:16
for a, b, k being ( ( ) ( ext-real V38() real ) Real) st k : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( (abs (a : ( ( ) ( ext-real V38() real ) Real) + b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) <= ((abs a : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) + (abs b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) & ((abs a : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) + (abs b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) <= (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) * (max ((abs a : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ,(abs b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( ) ( ext-real V38() real ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) & (abs (a : ( ( ) ( ext-real V38() real ) Real) + b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) <= (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) * (max ((abs a : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ,(abs b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( ) ( ext-real V38() real ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ;

theorem :: LPSPACE2:17
for a, b, k being ( ( ) ( ext-real V38() real ) Real) st a : ( ( ) ( ext-real V38() real ) Real) >= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ext-real V38() real ) Real) >= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & k : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
(max (a : ( ( ) ( ext-real V38() real ) Real) ,b : ( ( ) ( ext-real V38() real ) Real) )) : ( ( ) ( ext-real V38() real ) set ) to_power k : ( ( ) ( ext-real V38() real ) Real) : ( ( real ) ( ext-real V38() real ) set ) <= (a : ( ( ) ( ext-real V38() real ) Real) to_power k : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) + (b : ( ( ) ( ext-real V38() real ) Real) to_power k : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;

theorem :: LPSPACE2:18
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for a, b being ( ( ) ( ext-real V38() real ) Real) st b : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
((abs a : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) (#) ((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = (abs (a : ( ( ) ( ext-real V38() real ) Real) (#) f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power b : ( ( ) ( ext-real V38() real ) Real) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:19
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for a, b being ( ( ) ( ext-real V38() real ) Real) st a : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
(a : ( ( ) ( ext-real V38() real ) Real) to_power b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) (#) ((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = (a : ( ( ) ( ext-real V38() real ) Real) (#) (abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power b : ( ( ) ( ext-real V38() real ) Real) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:20
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( real ) ( ext-real V38() real ) number )
for E being ( ( ) ( ) set ) holds (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) | E : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( real ) ( ext-real V38() real ) number ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) to_power k : ( ( real ) ( ext-real V38() real ) number ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) | E : ( ( ) ( ) set ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:21
for a, b, k being ( ( ) ( ext-real V38() real ) Real) st k : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
(abs (a : ( ( ) ( ext-real V38() real ) Real) + b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) <= (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) * (((abs a : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) + ((abs b : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;

theorem :: LPSPACE2:22
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) holds
( (abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & (abs g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) + ((abs g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: LPSPACE2:23
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) holds
( X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) & X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) ) ;

theorem :: LPSPACE2:24
for X being ( ( non empty ) ( non empty ) set )
for k being ( ( ) ( ext-real V38() real ) Real) st k : ( ( ) ( ext-real V38() real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in (dom f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( ) ( ) 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
((abs (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) <= ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) (#) (((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) + ((abs g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( ) ( ext-real V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;

theorem :: LPSPACE2:25
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) holds
f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) ;

theorem :: LPSPACE2:26
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for a being ( ( ) ( ext-real V38() real ) Real)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) holds
a : ( ( ) ( ext-real V38() real ) Real) (#) f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) ;

theorem :: LPSPACE2:27
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) holds
f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) - g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) ;

theorem :: LPSPACE2:28
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) holds
abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset of ) ;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
cluster Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) -> non empty add-closed multi-closed ;
end;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
cluster RLSStruct(# (Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. (RLSp_PFunct X : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) ) : ( ( ) ( V52( RLSp_PFunct X : ( ( non empty ) ( non empty ) set ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) ) ) Element of the carrier of (RLSp_PFunct X : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| ((Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ,(RLSp_PFunct X : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) )) : ( ( Function-like V41([:(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) , Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V1() V4([:(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V5( Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like non empty total V41([:(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) , Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ (Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) , Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V5( Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like non empty total V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) , Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) -> strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func RLSp_LpFunct (M,k) -> ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) equals :: LPSPACE2:def 3
 RLSStruct(# (Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(In ((0. (RLSp_PFunct X : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( V52( RLSp_PFunct X : ( ( ) ( ) set ) : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of the carrier of (RLSp_PFunct X : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) )) : ( ( ) ( ) Element of Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ,(add| ((Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(RLSp_PFunct X : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty ) RLSStruct ) )) : ( ( Function-like V41([:(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V1() V4([:(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V5( Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) Function-like non empty total V41([:(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) Element of bool [:[:(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ (Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V5( Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) Function-like non empty total V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) ;
end;

begin

theorem :: LPSPACE2:29
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for v, u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b6 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:30
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for a being ( ( ) ( ext-real V38() real ) Real)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( ext-real V38() real ) Real) (#) f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V38() real ) Real) * u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b6 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:31
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
( u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ext-real non positive negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) * u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b5 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b5 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) = (X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) | (dom f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) & ex v, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st
( v : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & v : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ext-real non positive negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) * u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b5 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b5 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) = X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) & v : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func AlmostZeroLpFunctions (M,k) -> ( ( non empty ) ( non empty ) Subset of ) equals :: LPSPACE2:def 4
 { f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) where f is ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) & f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= X : ( ( ) ( ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
cluster AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) -> non empty add-closed multi-closed ;
end;

theorem :: LPSPACE2:32
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) holds
( 0. (RLSp_LpFunct (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( V52( RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) ) ) Element of the carrier of (RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) = X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) & 0. (RLSp_LpFunct (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( V52( RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) ) ) Element of the carrier of (RLSp_LpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) in AlmostZeroLpFunctions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func RLSp_AlmostZeroLpFunct (M,k) -> ( ( non empty ) ( non empty ) RLSStruct ) equals :: LPSPACE2:def 5
 RLSStruct(# (AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(In ((0. (RLSp_LpFunct (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) ) : ( ( ) ( V52( RLSp_LpFunct (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) ) ) Element of the carrier of (RLSp_LpFunct (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) )) : ( ( ) ( ) Element of AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ,(add| ((AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(RLSp_LpFunct (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) )) : ( ( Function-like V41([:(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V1() V4([:(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V5( AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) Function-like non empty total V41([:(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) Element of bool [:[:(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ (AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V5( AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) Function-like non empty total V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(AlmostZeroLpFunctions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) ;
end;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
cluster RLSp_LpFunct (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) -> non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

theorem :: LPSPACE2:33
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for v, u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_AlmostZeroLpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b6 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:34
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for a being ( ( ) ( ext-real V38() real ) Real)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( ext-real V38() real ) Real) (#) f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V38() real ) Real) * u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_AlmostZeroLpFunct (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b6 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func a.e-eq-class_Lp (f,M,k) -> ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) equals :: LPSPACE2:def 6
 { h : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) where h is ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( h : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) & f : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) a.e.= h : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

theorem :: LPSPACE2:35
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) holds
ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & dom f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) 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 V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: LPSPACE2:36
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for g, f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:37
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ;

theorem :: LPSPACE2:38
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) holds
f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:39
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for g, f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) <> {} : ( ( ) ( ) set ) & a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE2:40
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE2:41
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:42
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:43
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:44
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, f1, g, g1 being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & not a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) is empty & not a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) is empty & a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
a.e-eq-class_Lp ((f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp ((f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:45
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, f1, g, g1 being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
a.e-eq-class_Lp ((f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp ((f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:46
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for a being ( ( ) ( ext-real V38() real ) Real)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & dom f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) 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 V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & dom g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) 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 V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & not a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) is empty & a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
a.e-eq-class_Lp ((a : ( ( ) ( ext-real V38() real ) Real) (#) f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp ((a : ( ( ) ( ext-real V38() real ) Real) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:47
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for a being ( ( ) ( ext-real V38() real ) Real)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
a.e-eq-class_Lp ((a : ( ( ) ( ext-real V38() real ) Real) (#) f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp ((a : ( ( ) ( ext-real V38() real ) Real) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func CosetSet (M,k) -> ( ( non empty ) ( non empty ) Subset-Family of ) equals :: LPSPACE2:def 7
 { (a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) where f is ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) } ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func addCoset (M,k) -> ( ( Function-like V41([:(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4([:(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V5( CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty total V41([:(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) BinOp of CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) means :: LPSPACE2:def 8
 for A, B being ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) )
for a, b being ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st a : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in A : ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) & b : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in B : ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) holds
it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . (A : ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ,B : ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) : ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = a.e-eq-class_Lp ((a : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + b : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func zeroCoset (M,k) -> ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) equals :: LPSPACE2:def 9
 a.e-eq-class_Lp ((X : ( ( ) ( ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func lmultCoset (M,k) -> ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V5( CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty total V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of [:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) means :: LPSPACE2:def 10
 for z being ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) )
for A being ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) )
for f being ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in A : ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) holds
it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . (z : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ,A : ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) : ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = a.e-eq-class_Lp ((z : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) (#) f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func Pre-Lp-Space (M,k) -> ( ( strict ) ( strict ) RLSStruct ) means :: LPSPACE2:def 11
( the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) & the addF of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V41([: the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ( V1() V4([: the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V5( the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) Function-like V41([: the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = addCoset (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( Function-like V41([:(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4([:(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V5( CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty total V41([:(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) BinOp of CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) & 0. it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = zeroCoset (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ) Element of CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) & the Mult of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V5( the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = lmultCoset (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V5( CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty total V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of [:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) );
end;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
cluster Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( strict ) ( strict ) RLSStruct ) -> non empty strict ;
end;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
cluster Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) -> right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

begin

theorem :: LPSPACE2:48
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) ;

theorem :: LPSPACE2:49
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) holds
( Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) in REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) <= Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) ) ;

theorem :: LPSPACE2:50
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st ex x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st
( f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ;

theorem :: LPSPACE2:51
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( (abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ;

theorem :: LPSPACE2:52
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func Lp-Norm (M,k) -> ( ( Function-like V41( the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Function of the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) means :: LPSPACE2:def 12
 for x being ( ( ) ( ) Point of ( ( ) ( ) set ) ) ex f being ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st
( f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & ex r being ( ( ) ( ext-real V38() real ) Real) st
( r : ( ( ) ( ext-real V38() real ) Real) = Integral (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,((abs f : ( ( Function-like ) ( V1() V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V4(X : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( Function-like ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) & it : ( ( Function-like V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like total V41(X : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = r : ( ( ) ( ext-real V38() real ) Real) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) );
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ;
func Lp-Space (M,k) -> ( ( non empty ) ( non empty ) NORMSTR ) equals :: LPSPACE2:def 13
 NORMSTR(# the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , the ZeroF of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) Element of the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) ) , the addF of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( Function-like V41([: the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) ) ) ( V1() V4([: the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V5( the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) ) Function-like V41([: the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V5( the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) ) Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,(Lp-Norm (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( Function-like V41( the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Function of the carrier of (Pre-Lp-Space (M : ( ( Function-like V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) )) : ( ( strict ) ( strict ) RLSStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) #) : ( ( strict ) ( strict ) NORMSTR ) ;
end;

theorem :: LPSPACE2:53
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ex f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st
( f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) & ( for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
ex r being ( ( ) ( ext-real V38() real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( ext-real V38() real ) Real) & r : ( ( ) ( ext-real V38() real ) Real) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) & ||.x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = r : ( ( ) ( ext-real V38() real ) Real) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ) ) ;

theorem :: LPSPACE2:54
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for a being ( ( ) ( ext-real V38() real ) Real)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for x, y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ( f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) implies f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b7 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) & ( f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) implies a : ( ( ) ( ext-real V38() real ) Real) (#) f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in a : ( ( ) ( ext-real V38() real ) Real) * x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b7 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: LPSPACE2:55
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & ex r being ( ( ) ( ext-real V38() real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( ext-real V38() real ) Real) & r : ( ( ) ( ext-real V38() real ) Real) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) & ||.x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = r : ( ( ) ( ext-real V38() real ) Real) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ) ;

theorem :: LPSPACE2:56
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE2:57
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) a.e.= X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE2:58
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) holds Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs (X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) V5( INT : ( ( ) ( non empty V27() V156() V157() V158() V159() V160() V162() ) set ) ) Function-like V163() V164() V165() V166() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V5( RAT : ( ( ) ( non empty V27() V156() V157() V158() V159() V162() ) set ) ) Function-like V163() V164() V165() V166() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE2:59
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for m, n being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) + (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / n : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,n : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) holds
( f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: LPSPACE2:60
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for m, n being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) st (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) + (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / n : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,n : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) holds
ex r1 being ( ( ) ( ext-real V38() real ) Real) st
( r1 : ( ( ) ( ext-real V38() real ) Real) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) & ex r2 being ( ( ) ( ext-real V38() real ) Real) st
( r2 : ( ( ) ( ext-real V38() real ) Real) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power n : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) & Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) <= (r1 : ( ( ) ( ext-real V38() real ) Real) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) * (r2 : ( ( ) ( ext-real V38() real ) Real) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / n : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ) ;

theorem :: LPSPACE2:61
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,)
for m being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for r1, r2, r3 being ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) <= m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) & f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & r1 : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) & r2 : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) & r3 : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) + g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) holds
r3 : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) <= (r1 : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) + (r2 : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / m : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;

registration
let k be ( ( geq_than_1 ) ( non empty ext-real positive non negative V38() real geq_than_1 ) Real) ;
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
cluster Lp-Space (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( geq_than_1 ) ( non empty ext-real positive non negative V38() real geq_than_1 ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty ) NORMSTR ) -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ;
end;

begin

theorem :: LPSPACE2:62
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for Sq being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ex Fsq being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( Fsq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & Fsq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) & Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp ((Fsq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & ex r being ( ( ) ( ext-real V38() real ) Real) st
( r : ( ( ) ( ext-real V38() real ) Real) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs (Fsq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) & ||.(Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = r : ( ( ) ( ext-real V38() real ) Real) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ) ;

theorem :: LPSPACE2:63
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for Sq being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ex Fsq being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( Fsq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & Fsq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) & Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) = a.e-eq-class_Lp ((Fsq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & ex r being ( ( ) ( ext-real V38() real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( ext-real V38() real ) Real) & r : ( ( ) ( ext-real V38() real ) Real) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((abs (Fsq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() nonnegative ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V157() ) set ) ) & ||.(Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of (Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,b4 : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = r : ( ( ) ( ext-real V38() real ) Real) to_power (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) / k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( ) ( non empty ext-real positive non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ) ;

theorem :: LPSPACE2:64
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace)
for Sq being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) )
for Sq0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st ||.(Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) - Sq0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .|| : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is convergent & lim ||.(Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) - Sq0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .|| : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) = Sq0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE2:65
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace)
for Sq being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is Cauchy_sequence_by_Norm holds
ex N being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V167() ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V163() V164() V165() V166() V167() V169() ) sequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) st
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) st j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) >= N : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V167() ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V163() V164() V165() V166() V167() V169() ) sequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
||.((Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) . j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) - (Sq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) . (N : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V167() ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V163() V164() V165() V166() V167() V169() ) sequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real non negative V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) < 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) to_power (- i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real non positive V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;

theorem :: LPSPACE2:66
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for k being ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real)
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) holds F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) holds
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) holds (Partial_Sums F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) )) : ( ( ) ( functional non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) in Lp_Functions (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,k : ( ( positive ) ( non empty ext-real positive non negative V38() real ) Real) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ;

theorem :: LPSPACE2:67
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) holds F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) holds
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) holds (Partial_Sums F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) )) : ( ( ) ( functional non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: LPSPACE2:68
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) )
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) st F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) holds F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) & n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) holds
((Partial_Sums F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) )) : ( ( ) ( functional non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) <= ((Partial_Sums F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) )) : ( ( ) ( functional non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real ) Nat) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like V163() V164() V165() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real V38() real ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;

theorem :: LPSPACE2:69
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom holds
abs F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Functional_Sequence of ( ( ) ( functional non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) : ( ( ) ( functional non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) )) : ( ( ) ( functional non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is with_the_same_dom ;

theorem :: LPSPACE2:70
errorfrm ;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let k be ( ( geq_than_1 ) ( non empty ext-real positive non negative V38() real geq_than_1 ) Real) ;
cluster Lp-Space (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V157() ) set ) :] : ( ( ) ( non empty V164() ) set ) : ( ( ) ( non empty ) set ) ) ,k : ( ( geq_than_1 ) ( non empty ext-real positive non negative V38() real geq_than_1 ) Element of REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) : ( ( non empty ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like ) NORMSTR ) -> non empty complete ;
end;

begin

theorem :: LPSPACE2:71
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds CosetSet M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = CosetSet (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ;

theorem :: LPSPACE2:72
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds addCoset M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V41([:(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( V1() V4([:(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) V5( CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty total V41([:(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:[:(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = addCoset (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V41([:(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4([:(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V5( CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty total V41([:(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) BinOp of CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ;

theorem :: LPSPACE2:73
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds zeroCoset M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = zeroCoset (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ;

theorem :: LPSPACE2:74
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds lmultCoset M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) V5( CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty total V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,(CosetSet b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool (L1_Functions b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Element of bool the carrier of (RLSp_PFunct b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = lmultCoset (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V5( CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty total V41([:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of [:REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ,(CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ;

theorem :: LPSPACE2:75
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds Pre-L-Space M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) = Pre-Lp-Space (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) ;

theorem :: LPSPACE2:76
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds L-1-Norm M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V41( the carrier of (Pre-L-Space b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( the carrier of (Pre-L-Space b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( the carrier of (Pre-L-Space b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Element of bool [: the carrier of (Pre-L-Space b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) :] : ( ( ) ( non empty V163() V164() V165() ) set ) : ( ( ) ( non empty ) set ) ) = Lp-Norm (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V41( the carrier of (Pre-Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( strict ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ) ( V1() V4( the carrier of (Pre-Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( strict ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V5( REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) Function-like non empty total V41( the carrier of (Pre-Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( strict ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) V163() V164() V165() ) Function of the carrier of (Pre-Lp-Space (b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( strict ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) ) ;

theorem :: LPSPACE2:77
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds L-1-Space M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealNormSpace-like ) NORMSTR ) = Lp-Space (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V172() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V157() ) set ) ) Function-like non empty total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V157() ) set ) ) V164() V172() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() ) Element of bool REAL : ( ( ) ( non empty V27() V156() V157() V158() V162() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) NORMSTR ) ;