EUCLID

:: EUCLID_3 semantic presentation

begin

definition

let z be ( ( complex ) ( complex ) number ) ;

func cpx2euc z -> ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) equals :: EUCLID_3:def 1
|[(Re z : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ,(Im z : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ]| : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

end;

definition

let p be ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ;

func euc2cpx p -> ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) equals :: EUCLID_3:def 2
(p : ( ( ) ( ) RLTopStruct ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((p : ( ( ) ( ) RLTopStruct ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex ) set ) ;

end;

theorem :: EUCLID_3:1

for z being ( ( complex ) ( complex ) number ) holds euc2cpx (cpx2euc z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) = z : ( ( complex ) ( complex ) number ) ;

theorem :: EUCLID_3:2

theorem :: EUCLID_3:3

for z1, z2 being ( ( complex ) ( complex ) number ) st cpx2euc z1 : ( ( complex ) ( complex ) number ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = cpx2euc z2 : ( ( complex ) ( complex ) number ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) holds
z1 : ( ( complex ) ( complex ) number ) = z2 : ( ( complex ) ( complex ) number ) ;

theorem :: EUCLID_3:4

theorem :: EUCLID_3:5

for x1, x2 being ( ( ) ( complex V12() ext-real ) Real) holds cpx2euc (x1 : ( ( ) ( complex V12() ext-real ) Real) + (x2 : ( ( ) ( complex V12() ext-real ) Real) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex ) set ) ) : ( ( ) ( complex ) set ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = |[x1 : ( ( ) ( complex V12() ext-real ) Real) ,x2 : ( ( ) ( complex V12() ext-real ) Real) ]| : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_3:6

for z1, z2 being ( ( complex ) ( complex ) number ) holds |[(Re (z1 : ( ( complex ) ( complex ) number ) + z2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ,(Im (z1 : ( ( complex ) ( complex ) number ) + z2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ]| : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = |[((Re z1 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + (Re z2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ,((Im z1 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + (Im z2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ]| : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_3:7

for z1, z2 being ( ( complex ) ( complex ) number ) holds cpx2euc (z1 : ( ( complex ) ( complex ) number ) + z2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = (cpx2euc z1 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) + (cpx2euc z2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_3:8

theorem :: EUCLID_3:9

theorem :: EUCLID_3:10

for z being ( ( complex ) ( complex ) number ) holds |[(Re (- z : ( ( complex ) ( complex ) number ) ) : ( ( complex ) ( complex ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ,(Im (- z : ( ( complex ) ( complex ) number ) ) : ( ( complex ) ( complex ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ]| : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = |[(- (Re z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ,(- (Im z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ]| : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_3:11

for z being ( ( complex ) ( complex ) number ) holds cpx2euc (- z : ( ( complex ) ( complex ) number ) ) : ( ( complex ) ( complex ) set ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = - (cpx2euc z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_3:12

theorem :: EUCLID_3:13

theorem :: EUCLID_3:14

for z1, z2 being ( ( complex ) ( complex ) number ) holds cpx2euc (z1 : ( ( complex ) ( complex ) number ) - z2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = (cpx2euc z1 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - (cpx2euc z2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_3:15

theorem :: EUCLID_3:16

cpx2euc 0c : ( ( ) ( empty complex V129() V130() V131() V132() V133() V134() V135() ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = 0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_3:17

euc2cpx (0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) = 0c : ( ( ) ( empty complex V129() V130() V131() V132() V133() V134() V135() ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ;

theorem :: EUCLID_3:18

theorem :: EUCLID_3:19

for z being ( ( complex ) ( complex ) number )
for r being ( ( ) ( complex V12() ext-real ) Real) holds cpx2euc (r : ( ( ) ( complex V12() ext-real ) Real) * z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = r : ( ( ) ( complex V12() ext-real ) Real) * (cpx2euc z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_3:20

for r being ( ( ) ( complex V12() ext-real ) Real)
for p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) holds euc2cpx (r : ( ( ) ( complex V12() ext-real ) Real) * p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) = r : ( ( ) ( complex V12() ext-real ) Real) * (euc2cpx p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) : ( ( ) ( complex ) set ) ;

theorem :: EUCLID_3:21

theorem :: EUCLID_3:22

for f being ( ( ) ( Relation-like NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) -valued V21() FinSequence-like complex-yielding V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) st len f : ( ( ) ( Relation-like NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) -valued V21() FinSequence-like complex-yielding V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) = 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) holds
|.f : ( ( ) ( Relation-like NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) -valued V21() FinSequence-like complex-yielding V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = sqrt (((f : ( ( ) ( Relation-like NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) -valued V21() FinSequence-like complex-yielding V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) . 1 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ^2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((f : ( ( ) ( Relation-like NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) -valued V21() FinSequence-like complex-yielding V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) . 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ^2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ;

theorem :: EUCLID_3:23

for f being ( ( ) ( Relation-like NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) -valued V21() FinSequence-like complex-yielding V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) )
for p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st len f : ( ( ) ( Relation-like NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) -valued V21() FinSequence-like complex-yielding V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) = 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = f : ( ( ) ( Relation-like NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) -valued V21() FinSequence-like complex-yielding V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) holds
|.p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = |.f : ( ( ) ( Relation-like NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) -valued V21() FinSequence-like complex-yielding V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ;

theorem :: EUCLID_3:24

for z being ( ( complex ) ( complex ) number ) holds |.(cpx2euc z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = sqrt (((Re z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ^2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((Im z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ^2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ;

theorem :: EUCLID_3:25

definition

func Arg p -> ( ( ) ( complex V12() ext-real ) Real) equals :: EUCLID_3:def 3
Arg (euc2cpx p : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ;

end;

theorem :: EUCLID_3:26

for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) )
for p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st ( z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) = euc2cpx p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) or p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = cpx2euc z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) holds
Arg z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = Arg p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex V12() ext-real ) Real) ;

theorem :: EUCLID_3:27

for x1, x2 being ( ( ) ( complex V12() ext-real ) Real)
for p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( complex V12() ext-real ) Real) = |.p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (cos (Arg p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) & x2 : ( ( ) ( complex V12() ext-real ) Real) = |.p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (sin (Arg p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) holds
p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = |[x1 : ( ( ) ( complex V12() ext-real ) Real) ,x2 : ( ( ) ( complex V12() ext-real ) Real) ]| : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: EUCLID_3:28

Arg (0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex V12() ext-real ) Real) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_3:29

for p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> 0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
( ( Arg p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex V12() ext-real ) Real) < PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) implies Arg (- p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex V12() ext-real ) Real) = (Arg p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) + PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) & ( Arg p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex V12() ext-real ) Real) >= PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) implies Arg (- p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex V12() ext-real ) Real) = (Arg p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) - PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) ) ;

theorem :: EUCLID_3:30

theorem :: EUCLID_3:31

theorem :: EUCLID_3:32

theorem :: EUCLID_3:33

theorem :: EUCLID_3:34

definition

let p1, p2, p3 be ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ;

func angle (p1,p2,p3) -> ( ( ) ( complex V12() ext-real ) Real) equals :: EUCLID_3:def 4
angle ((euc2cpx p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ,(euc2cpx p2 : ( ( ) ( ) Element of p1 : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ,(euc2cpx p3 : ( ( V21() V30(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( V12() ) ( complex V12() ext-real ) set ) ;

end;

theorem :: EUCLID_3:35

theorem :: EUCLID_3:36

theorem :: EUCLID_3:37

theorem :: EUCLID_3:38

theorem :: EUCLID_3:39

for x, y being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) holds Re (x : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) .|. y : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = ((Re x : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (Re y : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((Im x : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (Im y : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ;

theorem :: EUCLID_3:40

for x, y being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) holds Im (x : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) .|. y : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = (- ((Re x : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (Im y : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((Im x : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (Re y : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V38() V129() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ;

theorem :: EUCLID_3:41

for p, q being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) holds |(p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) )| : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = ((p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (q : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (q : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ;

theorem :: EUCLID_3:42

theorem :: EUCLID_3:43

for p1, p2, p3 being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> 0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> 0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
( |(p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) )| : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) iff ( angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,(0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) or angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,(0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = (3 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty complex V12() ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) ) ;

theorem :: EUCLID_3:44

for p1, p2 being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> 0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> 0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
( ( ( not (- ((p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = |.p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * |.p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) & not (- ((p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = - (|.p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * |.p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real non positive ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) or angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,(0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) or angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,(0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = (3 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty complex V12() ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) & ( ( not angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,(0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) & not angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,(0. (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like zero complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = (3 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty complex V12() ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) or (- ((p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = |.p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * |.p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) or (- ((p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + ((p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = - (|.p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * |.p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real non positive ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) ) ;

theorem :: EUCLID_3:45

for p1, p2, p3 being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) & p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) holds
( |((p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) iff ( angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) or angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = (3 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty complex V12() ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) ) ;

theorem :: EUCLID_3:46

for p1, p2, p3 being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) & p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) & ( angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) or angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = (3 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty complex V12() ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) * PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ) holds
(|.(p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ^2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + (|.(p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ^2) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = |.(p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( complex V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ^2 : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ;

theorem :: EUCLID_3:47

for p1, p2, p3 being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) & p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) & angle (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) < PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) holds
((angle (p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex V12() ext-real ) Real) + (angle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex V12() ext-real ) Real) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + (angle (p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( complex V12() ext-real ) Real) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = PI : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) ;

definition

func Triangle (p1,p2,p3) -> ( ( ) ( ) Subset of ) equals :: EUCLID_3:def 5
((LSeg (p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ,p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg (p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg (p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

end;

definition

func closed_inside_of_triangle (p1,p2,p3) -> ( ( ) ( ) Subset of ) equals :: EUCLID_3:def 6
{ p : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) where p is ( ( ) ( Relation-like V21() V45(n : ( ( ) ( ) RLTopStruct ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( ) set ) ) : ex a1, a2, a3 being ( ( ) ( complex V12() ext-real ) Real) st
( 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) <= a1 : ( ( ) ( complex V12() ext-real ) Real) & 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) <= a2 : ( ( ) ( complex V12() ext-real ) Real) & 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) <= a3 : ( ( ) ( complex V12() ext-real ) Real) & (a1 : ( ( ) ( complex V12() ext-real ) Real) + a2 : ( ( ) ( complex V12() ext-real ) Real) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + a3 : ( ( ) ( complex V12() ext-real ) Real) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = 1 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = ((a1 : ( ( ) ( complex V12() ext-real ) Real) * p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (a2 : ( ( ) ( complex V12() ext-real ) Real) * p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (a3 : ( ( ) ( complex V12() ext-real ) Real) * p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) } ;

end;

definition

func inside_of_triangle (p1,p2,p3) -> ( ( ) ( ) Subset of ) equals :: EUCLID_3:def 7
(closed_inside_of_triangle (p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ,p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ) Subset of ) \ (Triangle (p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ,p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

end;

definition

func outside_of_triangle (p1,p2,p3) -> ( ( ) ( ) Subset of ) equals :: EUCLID_3:def 8
{ p : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) where p is ( ( ) ( Relation-like V21() V45(n : ( ( ) ( ) RLTopStruct ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( ) set ) ) : ex a1, a2, a3 being ( ( ) ( complex V12() ext-real ) Real) st
( ( 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) > a1 : ( ( ) ( complex V12() ext-real ) Real) or 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) > a2 : ( ( ) ( complex V12() ext-real ) Real) or 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) > a3 : ( ( ) ( complex V12() ext-real ) Real) ) & (a1 : ( ( ) ( complex V12() ext-real ) Real) + a2 : ( ( ) ( complex V12() ext-real ) Real) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + a3 : ( ( ) ( complex V12() ext-real ) Real) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = 1 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = ((a1 : ( ( ) ( complex V12() ext-real ) Real) * p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (a2 : ( ( ) ( complex V12() ext-real ) Real) * p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (a3 : ( ( ) ( complex V12() ext-real ) Real) * p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) } ;

end;

definition

func plane (p1,p2,p3) -> ( ( ) ( ) Subset of ) equals :: EUCLID_3:def 9
(outside_of_triangle (p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ,p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ) Subset of ) \/ (closed_inside_of_triangle (p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ,p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

end;

theorem :: EUCLID_3:48

theorem :: EUCLID_3:49

definition

pred q1,q2 are_lindependent2 means :: EUCLID_3:def 10
for a1, a2 being ( ( ) ( complex V12() ext-real ) Real) st (a1 : ( ( ) ( complex V12() ext-real ) Real) * q1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (a2 : ( ( ) ( complex V12() ext-real ) Real) * q2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = 0. (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( zero ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) holds
( a1 : ( ( ) ( complex V12() ext-real ) Real) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & a2 : ( ( ) ( complex V12() ext-real ) Real) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) );

end;

notation

end;

theorem :: EUCLID_3:50

theorem :: EUCLID_3:51

for n being ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) )
for p1, p2, p3, p0 being ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st p2 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) are_lindependent2 & p0 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) in plane (p1 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Subset of ) holds
ex a1, a2, a3 being ( ( ) ( complex V12() ext-real ) Real) st
( p0 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = ((a1 : ( ( ) ( complex V12() ext-real ) Real) * p1 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (a2 : ( ( ) ( complex V12() ext-real ) Real) * p2 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (a3 : ( ( ) ( complex V12() ext-real ) Real) * p3 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & (a1 : ( ( ) ( complex V12() ext-real ) Real) + a2 : ( ( ) ( complex V12() ext-real ) Real) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + a3 : ( ( ) ( complex V12() ext-real ) Real) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = 1 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for b1, b2, b3 being ( ( ) ( complex V12() ext-real ) Real) st p0 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) = ((b1 : ( ( ) ( complex V12() ext-real ) Real) * p1 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (b2 : ( ( ) ( complex V12() ext-real ) Real) * p2 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (b3 : ( ( ) ( complex V12() ext-real ) Real) * p3 : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL b₁ : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & (b1 : ( ( ) ( complex V12() ext-real ) Real) + b2 : ( ( ) ( complex V12() ext-real ) Real) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + b3 : ( ( ) ( complex V12() ext-real ) Real) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = 1 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( b1 : ( ( ) ( complex V12() ext-real ) Real) = a1 : ( ( ) ( complex V12() ext-real ) Real) & b2 : ( ( ) ( complex V12() ext-real ) Real) = a2 : ( ( ) ( complex V12() ext-real ) Real) & b3 : ( ( ) ( complex V12() ext-real ) Real) = a3 : ( ( ) ( complex V12() ext-real ) Real) ) ) ) ;

theorem :: EUCLID_3:52

theorem :: EUCLID_3:53

theorem :: EUCLID_3:54

definition

let n be ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let p1, p2, p3, p be ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ;
assume ( p2 : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) are_lindependent2 & p : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) in plane (p1 : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(n : ( ( ) ( natural complex V12() ext-real V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Subset of ) ) ;

func tricord1 (p1,p2,p3,p) -> ( ( ) ( complex V12() ext-real ) Real) means :: EUCLID_3:def 11
ex a2, a3 being ( ( ) ( complex V12() ext-real ) Real) st
( (it : ( ( V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( Relation-like p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + a2 : ( ( ) ( complex V12() ext-real ) Real) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + a3 : ( ( ) ( complex V12() ext-real ) Real) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = 1 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = ((it : ( ( V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( Relation-like p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (a2 : ( ( ) ( complex V12() ext-real ) Real) * p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (a3 : ( ( ) ( complex V12() ext-real ) Real) * p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) );

end;

definition

func tricord2 (p1,p2,p3,p) -> ( ( ) ( complex V12() ext-real ) Real) means :: EUCLID_3:def 12
ex a1, a3 being ( ( ) ( complex V12() ext-real ) Real) st
( (a1 : ( ( ) ( complex V12() ext-real ) Real) + it : ( ( V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( Relation-like p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + a3 : ( ( ) ( complex V12() ext-real ) Real) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = 1 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = ((a1 : ( ( ) ( complex V12() ext-real ) Real) * p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (it : ( ( V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( Relation-like p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (a3 : ( ( ) ( complex V12() ext-real ) Real) * p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) );

end;

definition

func tricord3 (p1,p2,p3,p) -> ( ( ) ( complex V12() ext-real ) Real) means :: EUCLID_3:def 13
ex a1, a2 being ( ( ) ( complex V12() ext-real ) Real) st
( (a1 : ( ( ) ( complex V12() ext-real ) Real) + a2 : ( ( ) ( complex V12() ext-real ) Real) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) + it : ( ( V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( Relation-like p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) = 1 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = ((a1 : ( ( ) ( complex V12() ext-real ) Real) * p1 : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (a2 : ( ( ) ( complex V12() ext-real ) Real) * p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) + (it : ( ( V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( Relation-like p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued V21() V30(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(p2 : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * p3 : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) );

end;

definition

func trcmap1 (p1,p2,p3) -> ( ( V21() V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) -valued V21() V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) ) complex-yielding V120() V121() ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V129() V130() V131() ) set ) ) means :: EUCLID_3:def 14
for p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) holds it : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) ) = tricord1 (p1 : ( ( ) ( ) RLTopStruct ) ,p2 : ( ( ) ( ) Element of p1 : ( ( ) ( ) RLTopStruct ) ) ,p3 : ( ( V21() V30(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) ;

end;

definition

func trcmap2 (p1,p2,p3) -> ( ( V21() V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) -valued V21() V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) ) complex-yielding V120() V121() ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V129() V130() V131() ) set ) ) means :: EUCLID_3:def 15
for p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) holds it : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) ) = tricord2 (p1 : ( ( ) ( ) RLTopStruct ) ,p2 : ( ( ) ( ) Element of p1 : ( ( ) ( ) RLTopStruct ) ) ,p3 : ( ( V21() V30(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) ;

end;

definition

func trcmap3 (p1,p2,p3) -> ( ( V21() V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) -valued V21() V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) ) complex-yielding V120() V121() ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V129() V130() V131() ) set ) ) means :: EUCLID_3:def 16
for p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) holds it : ( ( V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex V12() ext-real ) Element of the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V204() ) TopStruct ) : ( ( ) ( non empty V129() V130() V131() ) set ) ) = tricord3 (p1 : ( ( ) ( ) RLTopStruct ) ,p2 : ( ( ) ( ) Element of p1 : ( ( ) ( ) RLTopStruct ) ) ,p3 : ( ( V21() V30(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) ( Relation-like K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined p1 : ( ( ) ( ) RLTopStruct ) -valued V21() V30(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(p1 : ( ( ) ( ) RLTopStruct ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,p1 : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) ;

end;

theorem :: EUCLID_3:55

theorem :: EUCLID_3:56

for p1, p2, p3, p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) are_lindependent2 holds
( p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) in Triangle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Subset of ) iff ( tricord1 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) >= 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & tricord2 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) >= 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & tricord3 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) >= 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & ( tricord1 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) or tricord2 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) or tricord3 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ) ;

theorem :: EUCLID_3:57

for p1, p2, p3, p being ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) st p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) - p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V76() V141() V142() TopSpace-like V206() V207() V208() V209() V210() V211() V212() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) are_lindependent2 holds
( p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) in Triangle (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Subset of ) iff ( ( tricord1 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & tricord2 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) >= 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & tricord3 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) >= 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) or ( tricord1 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) >= 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & tricord2 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & tricord3 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) >= 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) or ( tricord1 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) >= 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & tricord2 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) >= 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) & tricord3 (p1 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( Relation-like V21() V45(2 : ( ( ) ( non empty natural complex V12() ext-real positive non negative V33() V34() V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V120() V121() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex V12() ext-real ) Real) = 0 : ( ( ) ( empty natural complex V12() ext-real non positive non negative V33() V34() V129() V130() V131() V132() V133() V134() V135() ) Element of NAT : ( ( ) ( V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V38() V129() V130() V131() V135() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ) ;

theorem :: EUCLID_3:58

theorem :: EUCLID_3:59