:: LPSPACE1 semantic presentation

begin

definition
let V be ( ( non empty ) ( non empty ) RLSStruct ) ;
let V1 be ( ( ) ( ) Subset of ) ;
attr V1 is multi-closed means :: LPSPACE1:def 1
 for a being ( ( ) ( V11() real ext-real ) Real)
for v being ( ( ) ( ) VECTOR of ( ( ) ( ) set ) ) st v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Element of V : ( ( ) ( ) NORMSTR ) ) holds
a : ( ( ) ( V11() real ext-real ) Real) * v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) in V1 : ( ( ) ( ) Element of V : ( ( ) ( ) NORMSTR ) ) ;
end;

theorem :: LPSPACE1:1
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RealLinearSpace)
for V1 being ( ( ) ( ) Subset of ) holds
( V1 : ( ( ) ( ) Subset of ) is linearly-closed iff ( V1 : ( ( ) ( ) Subset of ) is add-closed & V1 : ( ( ) ( ) Subset of ) is multi-closed ) ) ;

registration
let V be ( ( non empty ) ( non empty ) RLSStruct ) ;
cluster non empty add-closed multi-closed for ( ( ) ( ) Element of bool the carrier of V : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) RLSStruct ) ;
let X1 be ( ( non empty multi-closed ) ( non empty multi-closed ) Subset of ) ;
func Mult_ X1 -> ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X1 : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) RLSStruct ) ) :] : ( ( ) ( ) set ) ,X1 : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X1 : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) RLSStruct ) ) :] : ( ( ) ( ) set ) ) V17(X1 : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X1 : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) RLSStruct ) ) :] : ( ( ) ( ) set ) ,X1 : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X1 : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) RLSStruct ) ) :] : ( ( ) ( ) set ) ,X1 : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) equals :: LPSPACE1:def 2
 the Mult of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V17( the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) | [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X1 : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) RLSStruct ) ) :] : ( ( ) ( ) set ) : ( ( Function-like ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V17( the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) Function-like ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: LPSPACE1:2
for V being ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct )
for V1 being ( ( non empty ) ( non empty ) Subset of )
for d1 being ( ( ) ( ) Element of V1 : ( ( non empty ) ( non empty ) Subset of ) )
for A being ( ( Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) BinOp of V1 : ( ( non empty ) ( non empty ) Subset of ) )
for M being ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,V1 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,V1 : ( ( non empty ) ( non empty ) Subset of ) ) st d1 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ) ) = 0. V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( V55(b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) & A : ( ( Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ) ) = the addF of V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( Function-like V41([: the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) ( V13() V16([: the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V17( the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) Function-like V41([: the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || V1 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) set ) & M : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) = the Mult of V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V17( the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) | [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,V1 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( Function-like ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V17( the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) Function-like ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( RLSStruct(# V1 : ( ( non empty ) ( non empty ) Subset of ) ,d1 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,A : ( ( Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,M : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is Abelian & RLSStruct(# V1 : ( ( non empty ) ( non empty ) Subset of ) ,d1 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,A : ( ( Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,M : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is add-associative & RLSStruct(# V1 : ( ( non empty ) ( non empty ) Subset of ) ,d1 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,A : ( ( Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,M : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is right_zeroed & RLSStruct(# V1 : ( ( non empty ) ( non empty ) Subset of ) ,d1 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,A : ( ( Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,M : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is vector-distributive & RLSStruct(# V1 : ( ( non empty ) ( non empty ) Subset of ) ,d1 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,A : ( ( Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,M : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is scalar-distributive & RLSStruct(# V1 : ( ( non empty ) ( non empty ) Subset of ) ,d1 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,A : ( ( Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,M : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is scalar-associative & RLSStruct(# V1 : ( ( non empty ) ( non empty ) Subset of ) ,d1 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,A : ( ( Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:b2 : ( ( non empty ) ( non empty ) Subset of ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ) ) ,M : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is scalar-unital ) ;

theorem :: LPSPACE1:3
for V being ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct )
for V1 being ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) st 0. V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( V55(b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) in V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) holds
( RLSStruct(# V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) : ( ( ) ( V55(b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| (V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) )) : ( ( Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is Abelian & RLSStruct(# V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) : ( ( ) ( V55(b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| (V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) )) : ( ( Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is add-associative & RLSStruct(# V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) : ( ( ) ( V55(b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| (V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) )) : ( ( Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is right_zeroed & RLSStruct(# V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) : ( ( ) ( V55(b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| (V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) )) : ( ( Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is vector-distributive & RLSStruct(# V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) : ( ( ) ( V55(b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| (V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) )) : ( ( Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is scalar-distributive & RLSStruct(# V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) : ( ( ) ( V55(b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| (V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) )) : ( ( Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is scalar-associative & RLSStruct(# V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) : ( ( ) ( V55(b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| (V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,V : ( ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) )) : ( ( Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ V1 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is scalar-unital ) ;

theorem :: LPSPACE1:4
for V being ( ( non empty ) ( non empty ) RLSStruct )
for V1 being ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of )
for v, u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) )
for w1, w2 being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st w1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & w2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
w1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + w2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of RLSStruct(# b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. b1 : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( V55(b1 : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| (b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b1 : ( ( non empty ) ( non empty ) RLSStruct ) )) : ( ( Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE1:5
for V being ( ( non empty ) ( non empty ) RLSStruct )
for V1 being ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of )
for a being ( ( ) ( V11() real ext-real ) Real)
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) )
for w being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st w : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( V11() real ext-real ) Real) * w : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of RLSStruct(# b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,(In ((0. b1 : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( V55(b1 : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) )) : ( ( ) ( ) Element of b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ,(add| (b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b1 : ( ( non empty ) ( non empty ) RLSStruct ) )) : ( ( Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Element of bool [:[:b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17(b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty add-closed multi-closed ) ( non empty add-closed multi-closed ) Subset of ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( V11() real ext-real ) Real) * v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ;

begin

definition
let A, B be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Function-like V41([:(PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ) ) ( V13() V16([:(PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ) Function-like V41([:(PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ) ) BinOp of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ) ;
let f, g be ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) set ) ) V17(B : ( ( non empty ) ( non empty ) set ) ) Function-like ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ) ;
:: original: .
redefine func F . (f,g) -> ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17(B : ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Function-like ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,B : ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) RLSStruct ) ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) ,B : ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ) ;
end;

definition
let A be ( ( non empty ) ( non empty ) set ) ;
func multpfunc A -> ( ( Function-like V41([:(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) means :: LPSPACE1:def 3
 for f, g being ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds it : ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) RLSStruct ) ) . (f : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = f : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) (#) g : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;
end;

definition
let A be ( ( non empty ) ( non empty ) set ) ;
func multrealpfunc A -> ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) means :: LPSPACE1:def 4
 for a being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds it : ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) RLSStruct ) ) . (a : ( ( ) ( V11() real ext-real ) Real) ,f : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = a : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;
end;

definition
let A be ( ( non empty ) ( non empty ) set ) ;
func RealPFuncZero A -> ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) equals :: LPSPACE1:def 5
A : ( ( non empty ) ( non empty ) RLSStruct ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V41(A : ( ( non empty ) ( non empty ) RLSStruct ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like V41(A : ( ( non empty ) ( non empty ) RLSStruct ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) V173() V174() V175() V176() ) Element of bool [:A : ( ( non empty ) ( non empty ) RLSStruct ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) V173() V174() V175() V176() ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let A be ( ( non empty ) ( non empty ) set ) ;
func RealPFuncUnit A -> ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) equals :: LPSPACE1:def 6
A : ( ( non empty ) ( non empty ) RLSStruct ) --> 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V108() V166() V167() V168() V169() V170() V171() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V41(A : ( ( non empty ) ( non empty ) RLSStruct ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like V41(A : ( ( non empty ) ( non empty ) RLSStruct ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) V173() V174() V175() V176() ) Element of bool [:A : ( ( non empty ) ( non empty ) RLSStruct ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) V173() V174() V175() V176() ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: LPSPACE1:6
for A being ( ( non empty ) ( non empty ) set )
for h, f, g being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds
( h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) iff ( dom h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) set ) = (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) + (g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) ) ) ) ;

theorem :: LPSPACE1:7
for A being ( ( non empty ) ( non empty ) set )
for h, f, g being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds
( h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) iff ( dom h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) set ) = (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) * (g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) ) ) ) ;

theorem :: LPSPACE1:8
for A being ( ( non empty ) ( non empty ) set ) holds RealPFuncZero A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) <> RealPFuncUnit A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:9
for a being ( ( ) ( V11() real ext-real ) Real)
for A being ( ( non empty ) ( non empty ) set )
for h, f being ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds
( h : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (a : ( ( ) ( V11() real ext-real ) Real) ,f : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) iff ( dom h : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom f : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) in dom f : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
h : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) set ) = a : ( ( ) ( V11() real ext-real ) Real) * (f : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ) ;

theorem :: LPSPACE1:10
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:11
for A being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,((addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (((addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:12
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:13
for A being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,((multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (((multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:14
for A being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . ((RealPFuncUnit A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:15
for A being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . ((RealPFuncZero A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:16
for A being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,((multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V108() V166() V167() V168() V169() V170() V171() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (RealPFuncZero A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) | (dom f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE1:17
for A being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V108() V166() V167() V168() V169() V170() V171() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ,f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:18
for a, b being ( ( ) ( V11() real ext-real ) Real)
for A being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16(b3 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (a : ( ( ) ( V11() real ext-real ) Real) ,((multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (b : ( ( ) ( V11() real ext-real ) Real) ,f : ( ( ) ( V13() V16(b3 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . ((a : ( ( ) ( V11() real ext-real ) Real) * b : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,f : ( ( ) ( V13() V16(b3 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:19
for a, b being ( ( ) ( V11() real ext-real ) Real)
for A being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16(b3 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (((multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (a : ( ( ) ( V11() real ext-real ) Real) ,f : ( ( ) ( V13() V16(b3 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,((multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (b : ( ( ) ( V11() real ext-real ) Real) ,f : ( ( ) ( V13() V16(b3 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b3 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . ((a : ( ( ) ( V11() real ext-real ) Real) + b : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,f : ( ( ) ( V13() V16(b3 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of PFuncs (b3 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b3 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:20
for A being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,((addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (addpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (((multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,((multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (f : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,h : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

theorem :: LPSPACE1:21
for a being ( ( ) ( V11() real ext-real ) Real)
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) holds (multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (((multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (a : ( ( ) ( V11() real ext-real ) Real) ,f : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) = (multrealpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (a : ( ( ) ( V11() real ext-real ) Real) ,((multpfunc A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41([:(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) BinOp of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) . (f : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,g : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) )) : ( ( ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) : ( ( ) ( ) Element of PFuncs (b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of b2 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ;

definition
let A be ( ( non empty ) ( non empty ) set ) ;
func RLSp_PFunct A -> ( ( non empty ) ( non empty ) RLSStruct ) equals :: LPSPACE1:def 7
 RLSStruct(# (PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(RealPFuncZero A : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( V13() V16(A : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ,(addpfunc A : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( Function-like V41([:(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Element of bool [:[:(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(multrealpfunc A : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) ) V17( PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) :] : ( ( ) ( non empty ) set ) , PFuncs (A : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of A : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) ;
end;

registration
let A be ( ( non empty ) ( non empty ) set ) ;
cluster RLSp_PFunct A : ( ( non empty ) ( non empty ) set ) : ( ( non empty ) ( non empty ) RLSStruct ) -> non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

begin

theorem :: LPSPACE1:22
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V174() ) PartFunc of ,) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st E : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) = dom f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V174() ) 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 ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V174() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) = f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V174() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) set ) ) holds
( f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V174() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V174() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let r be ( ( ) ( V11() real ext-real ) Real) ;
:: original: -->
redefine func X --> r -> ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func L1_Functions M -> ( ( non empty ) ( non empty ) Subset of ) equals :: LPSPACE1:def 8
 { f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) where f is ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ex ND being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41([:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . ND : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) & dom f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( ) ( ) Element of bool X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) = ND : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ` : ( ( ) ( ) Element of bool X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41([:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

theorem :: LPSPACE1:23
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) holds
f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ;

theorem :: LPSPACE1:24
for a being ( ( ) ( V11() real ext-real ) Real)
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) holds
a : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) in L1_Functions M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
cluster L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) -> non empty add-closed multi-closed ;
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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func RLSp_L1Funct M -> ( ( non empty ) ( non empty ) RLSStruct ) equals :: LPSPACE1:def 9
 RLSStruct(# (L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(In ((0. (RLSp_PFunct X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( V55( RLSp_PFunct X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of the carrier of (RLSp_PFunct X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) )) : ( ( ) ( ) Element of L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ,(add| ((L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(RLSp_PFunct X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( non empty ) ( non empty ) RLSStruct ) )) : ( ( Function-like V41([:(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17( L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) Element of bool [:[:(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ (L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17( L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) #) : ( ( 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
cluster RLSp_L1Funct M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) RLSStruct ) -> non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

begin

theorem :: LPSPACE1:25
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,)
for v, u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_L1Funct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE1:26
for a being ( ( ) ( V11() real ext-real ) Real)
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,)
for u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( V11() real ext-real ) Real) * u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_L1Funct b4 : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f, g be ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ;
pred f a.e.= g,M means :: LPSPACE1:def 10
 ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st
( M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) . E : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) & f : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) | (E : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:X : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V41(X : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V41(X : ( ( non empty ) ( non empty ) RLSStruct ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V173() V174() V175() ) Element of bool [:X : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) | (E : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) `) : ( ( ) ( ) Element of bool X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:X : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) );
end;

theorem :: LPSPACE1:27
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,)
for u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
( u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V108() V166() V167() V168() V169() V170() V171() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) * u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (RLSp_L1Funct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_L1Funct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) = (X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) | (dom f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) & ex v, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st
( v : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & v : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V108() V166() V167() V168() V169() V170() V171() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) * u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (RLSp_L1Funct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_L1Funct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) & v : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) ;

theorem :: LPSPACE1:28
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) holds f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE1:29
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE1:30
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g, h being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= h : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= h : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE1:31
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, f1, g, g1 being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= f1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) a.e.= f1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + g1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE1:32
for a being ( ( ) ( V11() real ext-real ) Real)
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) holds
a : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) a.e.= a : ( ( ) ( V11() real ext-real ) Real) (#) g : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func AlmostZeroFunctions M -> ( ( non empty ) ( non empty ) Subset of ) equals :: LPSPACE1:def 11
 { f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) where f is ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) & f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= X : ( ( non empty ) ( non empty ) RLSStruct ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) ,M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

theorem :: LPSPACE1:33
for a being ( ( ) ( V11() real ext-real ) Real)
for X being ( ( non empty ) ( non empty ) set ) holds
( (X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) + (X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) = X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) & a : ( ( ) ( V11() real ext-real ) Real) (#) (X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) = X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
cluster AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) -> non empty add-closed multi-closed ;
end;

theorem :: LPSPACE1:34
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( 0. (RLSp_L1Funct M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( V55( RLSp_L1Funct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of (RLSp_L1Funct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) = X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) & 0. (RLSp_L1Funct M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( V55( RLSp_L1Funct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) ) ) Element of the carrier of (RLSp_L1Funct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) in AlmostZeroFunctions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func RLSp_AlmostZeroFunct M -> ( ( non empty ) ( non empty ) RLSStruct ) equals :: LPSPACE1:def 12
 RLSStruct(# (AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(In ((0. (RLSp_L1Funct M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( V55( RLSp_L1Funct M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of the carrier of (RLSp_L1Funct M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) )) : ( ( ) ( ) Element of AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ,(add| ((AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(RLSp_L1Funct M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) RLSStruct ) )) : ( ( Function-like V41([:(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17( AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) Element of bool [:[:(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ (AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ) V17( AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) , AlmostZeroFunctions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) ) #) : ( ( 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
cluster RLSp_L1Funct M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() ) RLSStruct ) -> non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

theorem :: LPSPACE1:35
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,)
for v, u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_AlmostZeroFunct b3 : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE1:36
for a being ( ( ) ( V11() real ext-real ) Real)
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,)
for u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( V11() real ext-real ) Real) * u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (RLSp_AlmostZeroFunct b4 : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ;
func a.e-eq-class (f,M) -> ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) equals :: LPSPACE1:def 13
 { g : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) where g is ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( g : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) & f : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) in L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) & f : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) a.e.= g : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

theorem :: LPSPACE1:37
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) holds
( g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in a.e-eq-class (f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE1:38
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) holds
f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in a.e-eq-class (f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE1:39
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) holds
( a.e-eq-class (f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class (g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) iff f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: LPSPACE1:40
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) holds
( a.e-eq-class (f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class (g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) iff g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in a.e-eq-class (f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE1:41
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, f1, g, g1 being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & f1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & a.e-eq-class (f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class (f1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & a.e-eq-class (g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class (g1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
a.e-eq-class ((f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class ((f1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + g1 : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: LPSPACE1:42
for a being ( ( ) ( V11() real ext-real ) Real)
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & a.e-eq-class (f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class (g : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
a.e-eq-class ((a : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = a.e-eq-class ((a : ( ( ) ( V11() real ext-real ) Real) (#) g : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func CosetSet M -> ( ( non empty ) ( non empty ) Subset-Family of ) equals :: LPSPACE1:def 14
 { (a.e-eq-class (f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) where f is ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func addCoset M -> ( ( Function-like V41([:(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V13() V16([:(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V17( CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like V41([:(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) BinOp of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) means :: LPSPACE1:def 15
 for A, B being ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) )
for a, b being ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st a : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in A : ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) & b : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in B : ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) holds
it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) : ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = a.e-eq-class ((a : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + b : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:X : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func zeroCoset M -> ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) means :: LPSPACE1:def 16
 ex f being ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st
( f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) = X : ( ( non empty ) ( non empty ) RLSStruct ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) & f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) & it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = a.e-eq-class (f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func lmultCoset M -> ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V17( CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) means :: LPSPACE1:def 17
 for z being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) )
for A being ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) )
for f being ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in A : ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) holds
it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . (z : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ,A : ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) : ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = a.e-eq-class ((z : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) (#) f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:X : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func Pre-L-Space M -> ( ( 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 V106() ) RLSStruct ) means :: LPSPACE1:def 18
( the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) & the addF of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V41([: the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ( V13() V16([: the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V17( the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) Function-like V41([: the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = addCoset M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V41([:(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V13() V16([:(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V17( CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like V41([:(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) BinOp of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) & 0. it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = zeroCoset M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) & the Mult of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V17( the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = lmultCoset M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) ) V17( CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of [:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,(CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( non empty ) set ) , CosetSet M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) );
end;

begin

theorem :: LPSPACE1:43
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) ;

theorem :: LPSPACE1:44
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) in REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) & Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) in REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) & abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: LPSPACE1:45
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) a.e.= abs g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) ) ;

theorem :: LPSPACE1:46
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st ex x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st
( f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) ) ;

theorem :: LPSPACE1:47
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: LPSPACE1:48
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) & Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func L-1-Norm M -> ( ( Function-like V41( the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ( V13() V16( the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V41( the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V173() V174() V175() ) Function of the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) means :: LPSPACE1:def 19
 for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ex f being ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st
( f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & it : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ) V17(X : ( ( non empty ) ( non empty ) RLSStruct ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) = Integral (M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ,(abs f : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(X : ( ( non empty ) ( non empty ) RLSStruct ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:X : ( ( non empty ) ( non empty ) RLSStruct ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) 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 V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func L-1-Space M -> ( ( non empty ) ( non empty ) NORMSTR ) equals :: LPSPACE1:def 20
 NORMSTR(# the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , the ZeroF of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( ) Element of the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) , the addF of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( Function-like V41([: the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) ( V13() V16([: the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V17( the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) Function-like V41([: the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) ( V13() V16([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V17( the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) Function-like V41([:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(L-1-Norm M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V41( the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ( V13() V16( the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V41( the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V173() V174() V175() ) Function of the carrier of (Pre-L-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) RLSStruct ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( 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 V106() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) #) : ( ( strict ) ( strict ) NORMSTR ) ;
end;

theorem :: LPSPACE1:49
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ex f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st
( f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty add-closed multi-closed ) Subset of ) & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = a.e-eq-class (f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & ||.x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) ) & ( for f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = ||.x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) ) ) ;

theorem :: LPSPACE1:50
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = a.e-eq-class (f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & ||.x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) = Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) ) ;

theorem :: LPSPACE1:51
for a being ( ( ) ( V11() real ext-real ) Real)
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,)
for x, y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ( f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) implies f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + g : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (L-1-Space b4 : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) & ( f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) implies a : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) in a : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (L-1-Space b4 : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: LPSPACE1:52
for r being ( ( ) ( V11() real ext-real ) Real)
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( ) ( ) Element of bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) : ( ( ) ( ) Element of bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) set ) = r : ( ( ) ( V11() real ext-real ) Real) ) holds
f : ( ( Function-like ) ( V13() V16(b2 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE1:53
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) in L1_Functions M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) a.e.= X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) ,M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE1:54
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs (X : ( ( non empty ) ( non empty ) set ) --> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) V17( RAT : ( ( ) ( non empty V29() V166() V167() V168() V169() V172() ) set ) ) V17( INT : ( ( ) ( non empty V29() V166() V167() V168() V169() V170() V172() ) set ) ) Function-like V173() V174() V175() V176() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of bool REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LPSPACE1:55
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs (f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) + g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) <= (Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) + (Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs g : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty ) ( non empty ) set ) ) V17( REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) ) Function-like V173() V174() V175() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V166() V167() V168() V172() ) set ) :] : ( ( ) ( non empty V173() V174() V175() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V167() ) set ) ) ;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() 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 V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
cluster L-1-Space M : ( ( Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V182() nonnegative sigma-additive ) ( V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V167() ) set ) ) Function-like V41(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V167() ) set ) ) V174() V182() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V167() ) set ) :] : ( ( ) ( non empty V174() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) NORMSTR ) -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealNormSpace-like ;
end;