:: EUCLID_6 semantic presentation

begin

theorem :: EUCLID_6:1
for p1, p2, p3, p4, p5, p6 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st sin (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = sin (angle (p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & cos (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = cos (angle (p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:2
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds sin (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = - (sin (angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:3
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds cos (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = cos (angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:4
for p1, p4, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (angle (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) or (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (angle (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ;

definition
let p1, p2, p3 be ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
func the_area_of_polygon3 (p1,p2,p3) -> ( ( real ) ( complex real ext-real ) number ) equals :: EUCLID_6:def 1
(((((p1 : ( ( ) ( ) MetrStruct ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (p2 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) -valued Function-like total V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - ((p2 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) -valued Function-like total V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (p1 : ( ( ) ( ) MetrStruct ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (((p2 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) -valued Function-like total V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (p3 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) MetrStruct ) -valued Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - ((p3 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) MetrStruct ) -valued Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (p2 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) -valued Function-like total V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (((p3 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) MetrStruct ) -valued Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (p1 : ( ( ) ( ) MetrStruct ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - ((p1 : ( ( ) ( ) MetrStruct ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (p3 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) MetrStruct ) -valued Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) / 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;
end;

definition
let p1, p2, p3 be ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
func the_perimeter_of_polygon3 (p1,p2,p3) -> ( ( real ) ( complex real ext-real ) number ) equals :: EUCLID_6:def 2
(|.(p2 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) -valued Function-like total V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) - p1 : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + |.(p3 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) MetrStruct ) -valued Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) - p2 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) -valued Function-like total V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + |.(p1 : ( ( ) ( ) MetrStruct ) - p3 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) MetrStruct ) -valued Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;
end;

theorem :: EUCLID_6:5
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds the_area_of_polygon3 (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( real ) ( complex real ext-real ) number ) = ((|.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (sin (angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) / 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:6
for p2, p1, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
|.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (sin (angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (sin (angle (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:7
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, c being ( ( real ) ( complex real ext-real ) number ) st a : ( ( real ) ( complex real ext-real ) number ) = |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & b : ( ( real ) ( complex real ext-real ) number ) = |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & c : ( ( real ) ( complex real ext-real ) number ) = |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
c : ( ( real ) ( complex real ext-real ) number ) ^2 : ( ( ) ( complex real ext-real ) set ) = ((a : ( ( real ) ( complex real ext-real ) number ) ^2) : ( ( ) ( complex real ext-real ) set ) + (b : ( ( real ) ( complex real ext-real ) number ) ^2) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) set ) - (((2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * (cos (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

begin

theorem :: EUCLID_6:8
for p, p1, p2 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:9
for p, p2, p3, p1 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:10
for p, p2, p3, p1 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:11
for p1, p, p2 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) ;

theorem :: EUCLID_6:12
for p, p1, p3, p4 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & not p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) holds
p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) ;

theorem :: EUCLID_6:13
for p, p1, p3, p2 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & not (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (angle (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
(angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (angle (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = 3 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:14
for p, p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & ( angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) / 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) or angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (3 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:15
for p, p1, p3, p2, p4 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:16
for p3, p1, p2 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:17
for p1, p2, p3, p being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( |((p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = 0 : ( ( ) ( empty natural complex real ext-real non positive non negative V33() V34() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) iff |((p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = 0 : ( ( ) ( empty natural complex real ext-real non positive non negative V33() V34() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: EUCLID_6:18
for p1, p3, p2, p being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & ( angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) / 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) or angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (3 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:19
for p1, p2, p3, p being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( ( angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) implies |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) & ( |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) implies |((p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = 0 : ( ( ) ( empty natural complex real ext-real non positive non negative V33() V34() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) & ( |((p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = 0 : ( ( ) ( empty natural complex real ext-real non positive non negative V33() V34() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) implies angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ) ;

definition
let p1, p2, p3 be ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
pred p1,p2,p3 is_collinear means :: EUCLID_6:def 3
( p1 : ( ( ) ( ) MetrStruct ) in LSeg (p2 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) -valued Function-like total V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,p3 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) MetrStruct ) -valued Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) or p2 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) -valued Function-like total V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) in LSeg (p3 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) MetrStruct ) -valued Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ,p1 : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) or p3 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) MetrStruct ) -valued Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) ) ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) in LSeg (p1 : ( ( ) ( ) MetrStruct ) ,p2 : ( ( Function-like V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ( Relation-like [:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) -valued Function-like total V30([:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:p1 : ( ( ) ( ) MetrStruct ) ,p1 : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) );
end;

notation
let p1, p2, p3 be ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
antonym p1,p2,p3 is_a_triangle for p1,p2,p3 is_collinear ;
end;

theorem :: EUCLID_6:20
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle iff ( p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) are_mutually_different & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) <> PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & angle (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) <> PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) <> PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ) ;

theorem :: EUCLID_6:21
for p1, p2, p3, p4, p5, p6 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
( |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ;

theorem :: EUCLID_6:22
for p1, p2, p3, p4, p5, p6 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
( |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ;

theorem :: EUCLID_6:23
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) are_mutually_different & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) <= PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
( angle (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) <= PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) <= PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ;

theorem :: EUCLID_6:24
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) are_mutually_different & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) > PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
( angle (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) > PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & angle (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) > PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ;

theorem :: EUCLID_6:25
for p, p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) is_a_triangle & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_6:26
for p, p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & not p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) <= PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
angle (p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) <= angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:27
for p, p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & not p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) > PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
angle (p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) >= angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:28
for p, p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & not p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) holds
ex p4 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st
( p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ;

theorem :: EUCLID_6:29
for p1, p2 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in inside_of_circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in outside_of_circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) holds
ex p being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in (LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) /\ (circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) )) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) ;

theorem :: EUCLID_6:30
for p1, p3, p4, p being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_6:31
for p1, p2, p, pc being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & pc : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = |[a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ]| : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & pc : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & not 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,pc : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * ((angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,pc : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:32
for p1 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & r : ( ( real ) ( complex real ext-real ) number ) > 0 : ( ( ) ( empty natural complex real ext-real non positive non negative V33() V34() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) holds
ex p2 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st
( p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & |[a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ]| : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_6:33
for p1, p2, p, pc being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & pc : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = |[a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ]| : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & not 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,pc : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * ((angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,pc : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:34
for p1, p2, p3, p4 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & not angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & not angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:35
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) <> PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:36
for p1, p2, p3, p4, p being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) are_mutually_different holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:37
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = 0 : ( ( ) ( empty natural complex real ext-real non positive non negative V33() V34() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) & p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_6:38
for p1, p2, p3, p4, p being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) holds
|.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

begin

theorem :: EUCLID_6:39
for p2, p1, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, s being ( ( real ) ( complex real ext-real ) number ) st a : ( ( real ) ( complex real ext-real ) number ) = |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & b : ( ( real ) ( complex real ext-real ) number ) = |.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & c : ( ( real ) ( complex real ext-real ) number ) = |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & s : ( ( real ) ( complex real ext-real ) number ) = (the_perimeter_of_polygon3 (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( real ) ( complex real ext-real ) number ) / 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) holds
abs (the_area_of_polygon3 (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = sqrt (((s : ( ( real ) ( complex real ext-real ) number ) * (s : ( ( real ) ( complex real ext-real ) number ) - a : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) set ) * (s : ( ( real ) ( complex real ext-real ) number ) - b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) set ) * (s : ( ( real ) ( complex real ext-real ) number ) - c : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) set ) : ( ( real ) ( complex real ext-real ) set ) ;

theorem :: EUCLID_6:40
for p1, p2, p3, p4, p being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for a, b, r being ( ( real ) ( complex real ext-real ) number ) st p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) & p : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V244( TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) holds
|.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (|.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (|.(p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * |.(p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

begin

theorem :: EUCLID_6:41
for p1, p2 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ;

theorem :: EUCLID_6:42
for p1, p2 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = 0 : ( ( ) ( empty natural complex real ext-real non positive non negative V33() V34() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) iff p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: EUCLID_6:43
for p1, p2 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.(p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:44
for p1, p2, p3, p4, p5, p6 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds not angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * (angle (p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:45
for p1, p2, p3, p4, p5, p6 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds not angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * (angle (p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (4 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:46
for p1, p2, p3, p4, p5, p6 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds not angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * (angle (p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (4 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:47
for p1, p2, p3, p4, p5, p6 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds not angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * (angle (p4 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (6 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:48
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for c1, c2 being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) st c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) = euc2cpx (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) & c2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) = euc2cpx (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) holds
angle (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ,c2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ;

theorem :: EUCLID_6:49
for c1, c2, c3 being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) holds
( (angle (c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ,c2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (angle (c2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ,c3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = angle (c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ,c3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) or (angle (c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ,c2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (angle (c2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ,c3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (angle (c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ,c3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ;

theorem :: EUCLID_6:50
for p1, p2, p3 being ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for c1, c2 being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) st c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) = euc2cpx (p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) & c2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) = euc2cpx (p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) holds
( Re (c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) .|. c2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (((p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * ((p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (((p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * ((p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & Im (c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) .|. c2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = (- (((p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * ((p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (((p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) * ((p3 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & |.c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = sqrt ((((p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) + (((p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) - (p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) & |.(p1 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) = |.c1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V126() V132() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) ;

theorem :: EUCLID_6:51
for n being ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) )
for q1 being ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( Function-like V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) -valued Function-like total V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V126() V127() V128() ) set ) ) st ( for q being ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) -valued Function-like total V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V126() V127() V128() ) set ) ) . q : ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) set ) = |.(q : ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - q1 : ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) holds
f : ( ( Function-like V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) -valued Function-like total V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V126() V127() V128() ) set ) ) is continuous ;

theorem :: EUCLID_6:52
for n being ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) )
for q1 being ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ex f being ( ( Function-like V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) -valued Function-like total V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V126() V127() V128() ) set ) ) st
( ( for q being ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) -valued Function-like total V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V126() V127() V128() ) set ) ) . q : ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) set ) = |.(q : ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - q1 : ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V42(b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) ) ) & f : ( ( Function-like V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) -valued Function-like total V30( the carrier of (TOP-REAL b1 : ( ( ) ( natural complex real ext-real V33() V34() V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) Element of bool REAL : ( ( ) ( non empty V35() V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V73() V138() V139() TopSpace-like V186() V187() V188() V189() V190() V191() V192() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V222() ) TopStruct ) : ( ( ) ( non empty V126() V127() V128() ) set ) ) complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V126() V127() V128() ) set ) ) is continuous ) ;