:: GROUP_12 semantic presentation

begin

registration
let I be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) ) ;
let i be ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ;
cluster F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like ) set ) . i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) -> Group-like for ( ( ) ( ) multMagma ) ;
end;

registration
let I be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) ) ;
let i be ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ;
cluster F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding associative ) set ) . i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) -> associative for ( ( ) ( ) multMagma ) ;
end;

registration
let I be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding commutative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding commutative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) ) ;
let i be ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ;
cluster F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding commutative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding commutative ) set ) . i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) -> commutative for ( ( ) ( ) multMagma ) ;
end;

theorem :: GROUP_12:1
for I being ( ( non empty ) ( non empty ) set )
for F being ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) )
for i being ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) )
for x being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function)
for g being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( ( dom x : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = I : ( ( non empty ) ( non empty ) set ) & x : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) = g : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & ( for j being ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) st j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
x : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) = 1_ (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) . j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( ) Element of the carrier of (b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) . b6 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ) iff x : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) = (1_ (product F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) +* (i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,g : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) ;

definition
let I be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) ) ;
let i be ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ;
func ProjSet (F,i) -> ( ( ) ( ) Subset of ) means :: GROUP_12:def 1
 for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in it : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) iff ex g being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) set ) = (1_ (product F : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) multMagma ) ) : ( ( ) ( ) Element of the carrier of (product F : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) ) +* (i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) ,g : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) );
end;

registration
let I be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) ) ;
let i be ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ;
cluster ProjSet (F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) ,i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Subset of ) -> non empty ;
end;

theorem :: GROUP_12:2
for I being ( ( non empty ) ( non empty ) set )
for F being ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) )
for i being ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) )
for x0 being ( ( ) ( ) set ) holds
( x0 : ( ( ) ( ) set ) in ProjSet (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ,i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) iff ex x being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ex g being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( x : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) = x0 : ( ( ) ( ) set ) & dom x : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = I : ( ( non empty ) ( non empty ) set ) & x : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) = g : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & ( for j being ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) st j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
x : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) = 1_ (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) . j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( ) Element of the carrier of (b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) . b7 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: GROUP_12:3
for I being ( ( non empty ) ( non empty ) set )
for F being ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) )
for i being ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) )
for g1, g2 being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) )
for z1, z2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st g1 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) = (1_ (product F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) +* (i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) & g2 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) = (1_ (product F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) +* (i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,z2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) holds
g1 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) * g2 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) = (1_ (product F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) +* (i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,(z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * z2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) . b3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;

theorem :: GROUP_12:4
for I being ( ( non empty ) ( non empty ) set )
for F being ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) )
for i being ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) )
for g1 being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) )
for z1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st g1 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) = (1_ (product F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) +* (i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) holds
g1 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) " : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) = (1_ (product F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) +* (i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,(z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ") : ( ( ) ( ) Element of the carrier of (b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) . b3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;

theorem :: GROUP_12:5
for I being ( ( non empty ) ( non empty ) set )
for F being ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) )
for i being ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) )
for g1, g2 being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) st g1 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) in ProjSet (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ,i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) & g2 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) in ProjSet (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ,i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) holds
g1 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) * g2 : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) in ProjSet (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ,i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) ;

theorem :: GROUP_12:6
for I being ( ( non empty ) ( non empty ) set )
for F being ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) )
for i being ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) )
for g being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) st g : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) in ProjSet (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ,i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) holds
g : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) " : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) in ProjSet (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ,i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) ;

definition
let I be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) ) ;
let i be ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ;
func ProjGroup (F,i) -> ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( ) ( ) set ) : ( ( strict ) ( strict ) multMagma ) ) means :: GROUP_12:def 2
 the carrier of it : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = ProjSet (F : ( ( ) ( ) set ) ,i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Subset of ) ;
end;

definition
let I be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) ) ;
let i be ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ;
func 1ProdHom (F,i) -> ( ( Function-like quasi_total V99(F : ( ( ) ( ) set ) . i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty ) multMagma ) , ProjGroup (F : ( ( ) ( ) set ) ,i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( ) ( ) set ) : ( ( strict ) ( strict ) multMagma ) ) ) ) ( Relation-like the carrier of (F : ( ( ) ( ) set ) . i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (ProjGroup (F : ( ( ) ( ) set ) ,i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( ) ( ) set ) : ( ( strict ) ( strict ) multMagma ) ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99(F : ( ( ) ( ) set ) . i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty ) multMagma ) , ProjGroup (F : ( ( ) ( ) set ) ,i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( ) ( ) set ) : ( ( strict ) ( strict ) multMagma ) ) ) ) Homomorphism of F : ( ( ) ( ) set ) . i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty ) multMagma ) , ProjGroup (F : ( ( ) ( ) set ) ,i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( ) ( ) set ) : ( ( strict ) ( strict ) multMagma ) ) ) means :: GROUP_12:def 3
 for x being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds it : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (ProjGroup (F : ( ( ) ( ) set ) ,i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( ) ( ) set ) : ( ( strict ) ( strict ) multMagma ) ) : ( ( ) ( non empty ) set ) ) = (1_ (product F : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) multMagma ) ) : ( ( ) ( ) Element of the carrier of (product F : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) ) +* (i : ( ( ) ( ) Element of I : ( ( ) ( ) set ) ) ,x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;
end;

registration
let I be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) ) ;
let i be ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ;
cluster 1ProdHom (F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) ,i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total V99(F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) . i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) , ProjGroup (F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) ,i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) ) ( Relation-like the carrier of (F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) . i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (ProjGroup (F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) ,i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99(F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) . i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) , ProjGroup (F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) ,i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) ) Homomorphism of F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) . i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) , ProjGroup (F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) ,i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) -> Function-like quasi_total bijective V99(F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) . i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) , ProjGroup (F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) ,i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) ;
end;

registration
let I be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) ) ;
let i be ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ;
cluster ProjGroup (F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) ,i : ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product F : ( ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like I : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) set ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) -> strict normal ;
end;

theorem :: GROUP_12:7
for I being ( ( non empty ) ( non empty ) set )
for F being ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of I : ( ( non empty ) ( non empty ) set ) )
for i, j being ( ( ) ( ) Element of I : ( ( non empty ) ( non empty ) set ) )
for x, y being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) st i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) in ProjGroup (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ,i : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative normal V153() V154() V155() V156() V157() V158() ) Subgroup of product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) & y : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) in ProjGroup (F : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ,j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative normal V153() V154() V155() V156() V157() V158() ) Subgroup of product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) holds
x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) = y : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) * x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) ;

theorem :: GROUP_12:8
for n being ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat)
for F being ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) )
for J being ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat)
for GJ being ( ( non empty Group-like associative ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) Group) st 1 : ( ( ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) <= J : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) & J : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) <= n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & GJ : ( ( non empty Group-like associative ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) Group) = F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) . J : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) : ( ( ) ( ) set ) holds
for x being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) )
for s being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) st len s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) < J : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) st k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) in dom s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) holds
s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) . k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( ) ( ) set ) in ProjGroup (F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ,k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative normal V153() V154() V155() V156() V157() V158() ) Subgroup of product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) & x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) = Product s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) holds
x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) . J : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) : ( ( ) ( ) set ) = 1_ GJ : ( ( non empty Group-like associative ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( ) Element of the carrier of b4 : ( ( non empty Group-like associative ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) ) ;

theorem :: GROUP_12:9
for n being ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat)
for F being ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) )
for x being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) )
for s being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) st len s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) . k : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) : ( ( ) ( ) set ) in ProjGroup (F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ,k : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) & x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) = Product s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) holds
for i being ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) st 1 : ( ( ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) & i : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) <= n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) holds
ex si being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) st
( si : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) = s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) . i : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) : ( ( ) ( ) set ) & x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) . i : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) : ( ( ) ( ) set ) = si : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) . i : ( ( V30() ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Nat) : ( ( ) ( ) set ) ) ;

theorem :: GROUP_12:10
for n being ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat)
for F being ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) )
for x being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) )
for s, t being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) st len s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) . k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( ) ( ) set ) in ProjGroup (F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ,k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative normal V153() V154() V155() V156() V157() V158() ) Subgroup of product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) & x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) = Product s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) & len t : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds t : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) . k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( ) ( ) set ) in ProjGroup (F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ,k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative normal V153() V154() V155() V156() V157() V158() ) Subgroup of product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) & x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) = Product t : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) holds
s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) = t : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) ;

theorem :: GROUP_12:11
for n being ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat)
for F being ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) )
for x being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) ex s being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) st
( len s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) . k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( ) ( ) set ) in ProjGroup (F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ,k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative normal V153() V154() V155() V156() V157() V158() ) Subgroup of product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) & x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) = Product s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: GROUP_12:12
for n being ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat)
for G being ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group)
for F being ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) st ( for i being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) . i : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) : ( ( non empty ) ( non empty ) multMagma ) is ( ( ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Subgroup of G : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) & ( for x being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex s being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) st
( len s : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds s : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) . k : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) in F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) . k : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) multMagma ) ) & x : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) = Product s : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) ) ) ) & ( for s, t being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) st len s : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) : ( ( ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds s : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) . k : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) in F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) . k : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) multMagma ) ) & len t : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds t : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) . k : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) in F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) . k : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) multMagma ) ) & Product s : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) ) = Product t : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) ) holds
s : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) = t : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) ) holds
ex f being ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) Homomorphism of product F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,G : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) st
( f : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) is bijective & ( for x being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) ex s being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) st
( len s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) . k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( ) ( ) set ) in F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) . k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( non empty ) ( non empty unital Group-like associative V153() V154() V155() V156() V157() V158() ) multMagma ) ) & s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ,b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) ) . x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) ) = Product s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty Group-like associative commutative ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) Group) : ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: GROUP_12:13
for n being ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat)
for G, F being ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) st ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) . k : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) : ( ( non empty ) ( non empty ) multMagma ) = ProjGroup (G : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ,k : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative V153() V154() V155() V156() V157() V158() ) Subgroup of product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) holds
ex f being ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) Homomorphism of product F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product G : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) st
( f : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) is bijective & ( for x being ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) ex s being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) st
( len s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() V25() V26() V30() V31() V33() V38() V100() ext-real non negative countable ) Element of NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) & ( for k being ( ( ) ( ) Element of Seg n : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) holds s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) . k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( ) ( ) set ) in F : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) . k : ( ( ) ( ) Element of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( non empty ) ( non empty unital Group-like associative commutative V153() V154() V155() V156() V157() V158() ) multMagma ) ) & s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) ( Relation-like the carrier of (product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V99( product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) ) Homomorphism of product b3 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) , product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) ) . x : ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) = Product s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V33() FinSequence-like FinSubsequence-like countable ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (product b2 : ( ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total multMagma-yielding Group-like associative commutative ) ( Relation-like Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) -defined Function-like total V178() multMagma-yielding Group-like associative commutative ) multMagma-Family of Seg b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty V30() ) ( non empty V24() V25() V26() V30() V31() V33() V38() V100() ext-real positive non negative countable ) Nat) ) countable ) Element of K19(NAT : ( ( ) ( non empty V24() V25() V26() V33() V38() V39() countable denumerable ) Element of K19(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33() ) set ) ) ) ) : ( ( strict ) ( non empty strict unital Group-like associative commutative V143() V153() V154() V155() V156() V157() V158() ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ) ) ;