:: MENELAUS semantic presentation

begin

notation
let X, Y be ( ( ) ( ) Subset of ) ;
synonym X is_parallel_to Y for X misses Y;
end;

definition
let X, Y, Z be ( ( ) ( ) Subset of ) ;
pred X,Y,Z are_concurrent means :: MENELAUS:def 1
( ( X : ( ( ) ( ) RLTopStruct ) is_parallel_to Y : ( ( ) ( ) Element of X : ( ( ) ( ) RLTopStruct ) ) & Y : ( ( ) ( ) Element of X : ( ( ) ( ) RLTopStruct ) ) is_parallel_to Z : ( ( V12() V21([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) ( V7() V10([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ) V11(X : ( ( ) ( ) RLTopStruct ) ) V12() V21([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) Element of bool [:[:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & Z : ( ( V12() V21([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) ( V7() V10([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ) V11(X : ( ( ) ( ) RLTopStruct ) ) V12() V21([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) Element of bool [:[:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) is_parallel_to X : ( ( ) ( ) RLTopStruct ) ) or ex A being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) st
( A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) RLTopStruct ) & A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Element of X : ( ( ) ( ) RLTopStruct ) ) & A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) in Z : ( ( V12() V21([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) ( V7() V10([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ) V11(X : ( ( ) ( ) RLTopStruct ) ) V12() V21([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) Element of bool [:[:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) );
end;

theorem :: MENELAUS:1
for A, B being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) holds
( (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) + B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) & (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) + B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) ;

theorem :: MENELAUS:2
for A being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda being ( ( ) ( ext-real V66() V67() ) Real) holds
( (lambda : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = lambda : ( ( ) ( ext-real V66() V67() ) Real) * (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) & (lambda : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = lambda : ( ( ) ( ext-real V66() V67() ) Real) * (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) ;

theorem :: MENELAUS:3
for A being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) holds
( (- A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = - (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) & (- A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = - (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) ;

theorem :: MENELAUS:4
for A, B being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda, mu being ( ( ) ( ext-real V66() V67() ) Real) holds
( ((lambda : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = (lambda : ( ( ) ( ext-real V66() V67() ) Real) * (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) & ((lambda : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = (lambda : ( ( ) ( ext-real V66() V67() ) Real) * (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) ;

theorem :: MENELAUS:5
for A being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda being ( ( ) ( ext-real V66() V67() ) Real) holds
( ((- lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = - (lambda : ( ( ) ( ext-real V66() V67() ) Real) * (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) & ((- lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = - (lambda : ( ( ) ( ext-real V66() V67() ) Real) * (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) ;

theorem :: MENELAUS:6
for A, B being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda, mu being ( ( ) ( ext-real V66() V67() ) Real) holds
( ((lambda : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) - (mu : ( ( ) ( ext-real V66() V67() ) Real) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = (lambda : ( ( ) ( ext-real V66() V67() ) Real) * (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) - (mu : ( ( ) ( ext-real V66() V67() ) Real) * (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) & ((lambda : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) - (mu : ( ( ) ( ext-real V66() V67() ) Real) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = (lambda : ( ( ) ( ext-real V66() V67() ) Real) * (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) - (mu : ( ( ) ( ext-real V66() V67() ) Real) * (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) ;

theorem :: MENELAUS:7
for A, A1, B, C being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda being ( ( ) ( ext-real V66() V67() ) Real) holds the_area_of_polygon3 ((((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( V67() ) ( ext-real V66() V67() ) set ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (the_area_of_polygon3 (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( V67() ) ( ext-real V66() V67() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * (the_area_of_polygon3 (A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( V67() ) ( ext-real V66() V67() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ;

theorem :: MENELAUS:8
for A, B, C being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) st angle (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = 0 : ( ( ) ( ext-real non positive non negative empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() V125() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) are_mutually_different & not angle (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = PI : ( ( ) ( ext-real positive non negative non empty V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) holds
angle (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = PI : ( ( ) ( ext-real positive non negative non empty V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ;

theorem :: MENELAUS:9
for A, B, C being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) holds
( A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear iff the_area_of_polygon3 (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( V67() ) ( ext-real V66() V67() ) set ) = 0 : ( ( ) ( ext-real non positive non negative empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() V125() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: MENELAUS:10
for B, C being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) holds the_area_of_polygon3 ((0. (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V47( TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( V67() ) ( ext-real V66() V67() ) set ) = (((B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) - ((C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) / 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ;

theorem :: MENELAUS:11
for A, A1, B, C being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) holds the_area_of_polygon3 ((A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) + A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( V67() ) ( ext-real V66() V67() ) set ) = ((the_area_of_polygon3 (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( V67() ) ( ext-real V66() V67() ) set ) + (the_area_of_polygon3 (A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( V67() ) ( ext-real V66() V67() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) - (the_area_of_polygon3 ((0. (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V47( TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( V67() ) ( ext-real V66() V67() ) set ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ;

theorem :: MENELAUS:12
for A, B, C being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) in Line (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MENELAUS:13
for B, C, A being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) st B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) <> C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) holds
( A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear iff A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) in Line (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MENELAUS:14
for A, B, C, A1 being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda being ( ( ) ( ext-real V66() V67() ) Real) st A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) <> A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: MENELAUS:15
for A, B, C being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle holds
( A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle ) ;

theorem :: MENELAUS:16
for A, B, C, A1, B1 being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda, mu being ( ( ) ( ext-real V66() V67() ) Real) st A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & mu : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) <> 0 : ( ( ) ( ext-real non positive non negative empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() V125() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) iff not Line (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is_parallel_to Line (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

begin

theorem :: MENELAUS:17
for A1, B, C, B1, A, C1 being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda, mu, nu being ( ( ) ( ext-real V66() V67() ) Real) st A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (nu : ( ( ) ( ext-real V66() V67() ) Real) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
the_area_of_polygon3 (A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( V67() ) ( ext-real V66() V67() ) set ) = ((((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + ((lambda : ( ( ) ( ext-real V66() V67() ) Real) * mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (the_area_of_polygon3 (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( V67() ) ( ext-real V66() V67() ) set ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ;

theorem :: MENELAUS:18
for A, B, C, A1, B1, C1 being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda, mu, nu being ( ( ) ( ext-real V66() V67() ) Real) st A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (nu : ( ( ) ( ext-real V66() V67() ) Real) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & lambda : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & mu : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & nu : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear iff ((lambda : ( ( ) ( ext-real V66() V67() ) Real) / (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (mu : ( ( ) ( ext-real V66() V67() ) Real) / (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (nu : ( ( ) ( ext-real V66() V67() ) Real) / (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = - 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real non positive negative non empty V66() V67() V68() V69() ) Element of K113() : ( ( ) ( non empty V31() V119() V120() V121() V122() V123() V125() ) set ) ) ) ;

theorem :: MENELAUS:19
for A, B, C, A1, B1, C1, C2, A2, B2 being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda, mu, nu being ( ( ) ( ext-real V66() V67() ) Real) st A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (nu : ( ( ) ( ext-real V66() V67() ) Real) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & lambda : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & mu : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & nu : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear holds
( (((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (nu : ( ( ) ( ext-real V66() V67() ) Real) * lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) <> 0 : ( ( ) ( ext-real non positive non negative empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() V125() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & the_area_of_polygon3 (A2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( V67() ) ( ext-real V66() V67() ) set ) = (((((mu : ( ( ) ( ext-real V66() V67() ) Real) * nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) - (((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ^2) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) / ((((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (nu : ( ( ) ( ext-real V66() V67() ) Real) * lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (the_area_of_polygon3 (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( V67() ) ( ext-real V66() V67() ) set ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) ;

theorem :: MENELAUS:20
for A, B, C, A1, B1, C1, C2, A2, B2 being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) / 3 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real positive non negative non empty V66() V67() V69() ) Element of K112() : ( ( ) ( non empty V31() V119() V120() V121() V122() V125() ) set ) ) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) / 3 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real positive non negative non empty V66() V67() V69() ) Element of K112() : ( ( ) ( non empty V31() V119() V120() V121() V122() V125() ) set ) ) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) / 3 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real positive non negative non empty V66() V67() V69() ) Element of K112() : ( ( ) ( non empty V31() V119() V120() V121() V122() V125() ) set ) ) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) / 3 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real positive non negative non empty V66() V67() V69() ) Element of K112() : ( ( ) ( non empty V31() V119() V120() V121() V122() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) / 3 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real positive non negative non empty V66() V67() V69() ) Element of K112() : ( ( ) ( non empty V31() V119() V120() V121() V122() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) / 3 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real positive non negative non empty V66() V67() V69() ) Element of K112() : ( ( ) ( non empty V31() V119() V120() V121() V122() V125() ) set ) ) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear holds
the_area_of_polygon3 (A2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( V67() ) ( ext-real V66() V67() ) set ) = (the_area_of_polygon3 (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( V67() ) ( ext-real V66() V67() ) set ) / 7 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ;

theorem :: MENELAUS:21
for A, B, C, A1, B1, C1 being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda, mu, nu being ( ( ) ( ext-real V66() V67() ) Real) st A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (nu : ( ( ) ( ext-real V66() V67() ) Real) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & lambda : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & mu : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & nu : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) <> 0 : ( ( ) ( ext-real non positive non negative empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() V125() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (nu : ( ( ) ( ext-real V66() V67() ) Real) * lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) <> 0 : ( ( ) ( ext-real non positive non negative empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() V125() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) <> 0 : ( ( ) ( ext-real non positive non negative empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() V125() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( ((lambda : ( ( ) ( ext-real V66() V67() ) Real) / (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (mu : ( ( ) ( ext-real V66() V67() ) Real) / (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (nu : ( ( ) ( ext-real V66() V67() ) Real) / (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) iff ex A2 being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) st
( A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear & C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A2 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_collinear ) ) ;

theorem :: MENELAUS:22
for A, B, C, A1, B1, C1 being ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) )
for lambda, mu, nu being ( ( ) ( ext-real V66() V67() ) Real) st A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (lambda : ( ( ) ( ext-real V66() V67() ) Real) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (mu : ( ( ) ( ext-real V66() V67() ) Real) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (nu : ( ( ) ( ext-real V66() V67() ) Real) * B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & lambda : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & mu : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) & nu : ( ( ) ( ext-real V66() V67() ) Real) <> 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( ((lambda : ( ( ) ( ext-real V66() V67() ) Real) / (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - lambda : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (mu : ( ( ) ( ext-real V66() V67() ) Real) / (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - mu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) * (nu : ( ( ) ( ext-real V66() V67() ) Real) / (1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) - nu : ( ( ) ( ext-real V66() V67() ) Real) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) : ( ( ) ( ext-real V66() V67() ) Element of K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) iff Line (A : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,A1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , Line (B : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,B1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , Line (C : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ,C1 : ( ( ) ( V7() V12() V38(2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) V70() V109() V110() V111() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the U1 of (TOP-REAL 2 : ( ( ) ( ext-real positive non negative non empty V30() V66() V67() V68() V69() V119() V120() V121() V122() V123() V124() ) Element of K114() : ( ( ) ( V119() V120() V121() V122() V123() V124() V125() ) Element of bool K110() : ( ( ) ( non empty V31() V119() V120() V121() V125() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V85() V146() V147() V148() V149() V150() V151() V152() strict V203() V204() ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) are_concurrent ) ;