:: EUCLID_8 semantic presentation

begin

definition
let x, y, z be ( ( real ) ( real ) number ) ;
:: original: <*
redefine func |[x,y,z]| -> ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;
end;

theorem :: EUCLID_8:1
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:2
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) FinSequence of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) st len f : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) FinSequence of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) = 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) FinSequence of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) is ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

definition
func <e1> -> ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) equals :: EUCLID_8:def 1
|[1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;
func <e2> -> ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) equals :: EUCLID_8:def 2
|[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;
func <e3> -> ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) equals :: EUCLID_8:def 3
|[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;
end;

definition
let p1, p2 be ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;
func p1 <X> p2 -> ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) equals :: EUCLID_8:def 4
|[(((p1 : ( ( ) ( ) set ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p2 : ( ( ) ( ) set ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((p1 : ( ( ) ( ) set ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p2 : ( ( ) ( ) set ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((p1 : ( ( ) ( ) set ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p2 : ( ( ) ( ) set ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((p1 : ( ( ) ( ) set ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p2 : ( ( ) ( ) set ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((p1 : ( ( ) ( ) set ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p2 : ( ( ) ( ) set ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((p1 : ( ( ) ( ) set ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p2 : ( ( ) ( ) set ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;
end;

theorem :: EUCLID_8:3
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) st p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) are_ldependent2 holds
p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = 0.REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of the U1 of (TOP-REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V202() ) ( V51() V77() V145() V146() V163() V190() V191() V192() V193() V194() V195() V196() V202() ) L16()) : ( ( ) ( ) set ) ) ;

begin

theorem :: EUCLID_8:4
|.<e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) .| : ( ( ) ( real V33() ext-real non negative ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:5
|.<e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) .| : ( ( ) ( real V33() ext-real non negative ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:6
|.<e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) .| : ( ( ) ( real V33() ext-real non negative ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:7
<e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) , <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) are_orthogonal ;

theorem :: EUCLID_8:8
<e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) , <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) are_orthogonal ;

theorem :: EUCLID_8:9
<e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) , <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) are_orthogonal ;

theorem :: EUCLID_8:10
|(<e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,<e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:11
|(<e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,<e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:12
|(<e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,<e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:13
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,|[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:14
canceled;

theorem :: EUCLID_8:15
canceled;

theorem :: EUCLID_8:16
<e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:17
<e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:18
<e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:19
<e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = - <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:20
<e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = - <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:21
<e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = - <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:22
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> |[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:23
canceled;

theorem :: EUCLID_8:24
canceled;

theorem :: EUCLID_8:25
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:26
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:27
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:28
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:29
canceled;

theorem :: EUCLID_8:30
canceled;

theorem :: EUCLID_8:31
- <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[(- 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real V33() V34() V35() ext-real ) Element of INT : ( ( ) ( V11() V38() V133() V134() V135() V136() V137() V139() ) set ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:32
- <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,(- 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real V33() V34() V35() ext-real ) Element of INT : ( ( ) ( V11() V38() V133() V134() V135() V136() V137() V139() ) set ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:33
- <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,(- 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real V33() V34() V35() ext-real ) Element of INT : ( ( ) ( V11() V38() V133() V134() V135() V136() V137() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:34
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:35
canceled;

theorem :: EUCLID_8:36
canceled;

theorem :: EUCLID_8:37
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:38
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:39
for x, y, z being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds |[x : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,z : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = ((x : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (y : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (z : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:40
for r, x, y, z being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |[x : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,z : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * x : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * y : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * z : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:41
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds - p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = ((- ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:42
for x, y, z being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds - |[x : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,z : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = ((- (x : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - (y : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - (z : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:43
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = ((((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:44
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = ((((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:45
for x1, x2, x3, y1, y2, y3 being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds |[x1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,x2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,x3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + |[y1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (((x1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + y1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((x2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + y2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((x3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + y3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:46
for x1, x2, x3, y1, y2, y3 being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds |[x1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,x2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,x3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - |[y1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (((x1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - y1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((x2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - y2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((x3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - y3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:47
for p1, p2, p3 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds
( (((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = ((((p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (((p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (((p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) iff (((p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = ((((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ;

definition
let f1, f2, f3 be ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ;
func VFunc (f1,f2,f3) -> ( ( V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ( Relation-like V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) Function of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) means :: EUCLID_8:def 5
 for t being ( ( ) ( real V33() ext-real ) Real) holds it : ( ( ) ( ) set ) . t : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[(f1 : ( ( ) ( ) set ) . t : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(f2 : ( ( ) ( ) set ) . t : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(f3 : ( ( ) ( ) set ) . t : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;
end;

theorem :: EUCLID_8:48
for f1, f2, f3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t being ( ( ) ( real V33() ext-real ) Real) holds (VFunc (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) )) : ( ( V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ( Relation-like V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) Function of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . t : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:49
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) )
for f1, f2, f3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t being ( ( ) ( real V33() ext-real ) Real) holds
( p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (VFunc (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) )) : ( ( V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ( Relation-like V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) Function of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . t : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) iff ( p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ;

theorem :: EUCLID_8:50
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds
( len p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) = 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & dom p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( V133() V134() V135() V136() V137() V138() ) Element of K19(NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) = Seg 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) V133() V134() V135() V136() V137() V138() ) Element of K19(NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:51
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds mlt (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) FinSequence of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = <*((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) set ) ;

theorem :: EUCLID_8:52
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:53
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds - p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[(- (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(- (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(- (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:54
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds len (- p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) = len p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:55
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:56
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) )
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t1, t2 being ( ( ) ( real V33() ext-real ) Real) st p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (VFunc (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) )) : ( ( V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ( Relation-like V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) Function of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . t1 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) & q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (VFunc (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) )) : ( ( V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ( Relation-like V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) Function of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . t2 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) & p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds
( f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t1 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t2 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t1 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t2 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t1 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t2 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:57
for f1, g1, f2, g2, f3, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t1, t2 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t1 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t2 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t1 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t2 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t1 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t2 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds
(VFunc (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) )) : ( ( V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ( Relation-like V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) Function of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . t1 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (VFunc (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) )) : ( ( V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ( Relation-like V6() V26( REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) Function of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) , REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . t2 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:58
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) )
for r being ( ( real ) ( real ) number ) holds r : ( ( real ) ( real ) number ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[(r : ( ( real ) ( real ) number ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(r : ( ( real ) ( real ) number ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(r : ( ( real ) ( real ) number ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:59
for x, y, z being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for r being ( ( real ) ( real ) number ) holds r : ( ( real ) ( real ) number ) * |[x : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,z : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[(r : ( ( real ) ( real ) number ) * x : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(r : ( ( real ) ( real ) number ) * y : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(r : ( ( real ) ( real ) number ) * z : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:60
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds - p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[(- (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(- (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(- (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:61
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds
( (- p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = - (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & (- p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = - (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & (- p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = - (p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:62
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:63
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = (((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:64
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = (((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:65
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |.p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) .| : ( ( ) ( real V33() ext-real non negative ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = sqrt ((((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:66
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |.(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) .| : ( ( ) ( real V33() ext-real non negative ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = (abs r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (sqrt ((((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:67
for r1, r2 being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds (r1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (r2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (r1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + r2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |[(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:68
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:69
for r1, r2 being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds (r1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - (r2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (r1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - r2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |[(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:70
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * ((((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + ((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) . 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:71
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,(0.REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of the U1 of (TOP-REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V202() ) ( V51() V77() V145() V146() V163() V190() V191() V192() V193() V194() V195() V196() V202() ) L16()) : ( ( ) ( ) set ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:72
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((- p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = - |(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:73
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((- p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,(- p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = |(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:74
for p1, p2, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = |(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - |(p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:75
for p1, p2, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = |(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + |(p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:76
for r1, r2 being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for p1, p2, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |(((r1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (r2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = (r1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (r2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |(p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:77
for p1, p2, q1, q2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,(q1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + q2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = ((|(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + |(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + |(p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + |(p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:78
for p1, p2, q1, q2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,(q1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - q2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = ((|(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - |(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - |(p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + |(p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:79
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds
( |(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) iff p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = 0.REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of the U1 of (TOP-REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V202() ) ( V51() V77() V145() V146() V163() V190() V191() V192() V193() V194() V195() V196() V202() ) L16()) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:80
for p being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds
( |.p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) .| : ( ( ) ( real V33() ext-real non negative ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) iff p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = 0.REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of the U1 of (TOP-REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V202() ) ( V51() V77() V145() V146() V163() V190() V191() V192() V193() V194() V195() V196() V202() ) L16()) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:81
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = (|(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) * |(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + |(q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:82
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |((p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) = (|(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + (2 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) * |(p : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) + |(q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:83
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = - (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:84
for x1, x2, x3, y1, y2, y3 being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds |[x1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,x2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,x3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> |[y1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,y3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[((x2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * y3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (x3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * y2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((x3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * y1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (x1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * y3 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((x1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * y2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (x2 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * y1 : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:85
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds
( (r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) & (r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> (r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) ;

theorem :: EUCLID_8:86
for p1, p2, p3 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:87
for p1, p2, p3 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds (p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:88
for p1, p2, q1, q2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds (p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> (q1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + q2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> q1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> q2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> q1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> q2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:89
for p1, p2, p3 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> (p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (|(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - (|(p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

definition
let p1, p2, p3 be ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;
func |{p1,p2,p3}| -> ( ( real ) ( real ) number ) equals :: EUCLID_8:def 6
|(p1 : ( ( ) ( ) set ) ,(p2 : ( ( ) ( ) set ) <X> p3 : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;
end;

theorem :: EUCLID_8:90
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |{p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) }| : ( ( real ) ( real ) number ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:91
for p2, p1 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |{p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) }| : ( ( real ) ( real ) number ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:92
for p1, p2 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |{p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) }| : ( ( real ) ( real ) number ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: EUCLID_8:93
for p1, p2, p3 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |{p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) }| : ( ( real ) ( real ) number ) = |{p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) }| : ( ( real ) ( real ) number ) ;

theorem :: EUCLID_8:94
for p1, p2, p3 being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |{p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) }| : ( ( real ) ( real ) number ) = |((p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p3 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

theorem :: EUCLID_8:95
for p1, p2, q being ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) holds |{p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) }| : ( ( real ) ( real ) number ) = |((q : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) <X> p1 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ,p2 : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) )| : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ;

begin

definition
let f1, f2, f3 be ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ;
let t0 be ( ( ) ( real V33() ext-real ) Real) ;
func VFuncdiff (f1,f2,f3,t0) -> ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) equals :: EUCLID_8:def 7
|[(diff (f1 : ( ( ) ( ) set ) ,t0 : ( ( ) ( ) set ) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(diff (f2 : ( ( ) ( ) set ) ,t0 : ( ( ) ( ) set ) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(diff (f3 : ( ( ) ( ) set ) ,t0 : ( ( ) ( ) set ) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;
end;

theorem :: EUCLID_8:96
for f1, f2, f3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:97
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) + g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) + g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) + g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (VFuncdiff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (VFuncdiff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:98
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) - g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) - g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) - g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (VFuncdiff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - (VFuncdiff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:99
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for f1, f2, f3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (VFuncdiff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:100
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[((g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + |[((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:101
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds
VFuncdiff ((g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) * f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) * f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) * f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[((diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,(f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,(f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,(f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:102
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) holds
VFuncdiff ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) / g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) / g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) / g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[((((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:103
for f1, f2, f3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) holds
VFuncdiff ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ^) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ^) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ^) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = - |[((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:104
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for f1, f2, f3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e1> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e2> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * <e3> : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:105
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) + g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) + g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) + g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (VFuncdiff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (VFuncdiff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:106
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff ((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) - g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) - g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) - g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (VFuncdiff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) - (r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (VFuncdiff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:107
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff (((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |[((g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + (r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |[((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:108
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) holds
VFuncdiff (((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) * f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |[((diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,(f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,(f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,(f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:109
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) holds
VFuncdiff (((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) / g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) / g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) / g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |[((((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:110
for r being ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) )
for f1, f2, f3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) & r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) <> 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( V11() V38() V133() V134() V135() V136() V139() ) set ) -valued V9() V11() ordinal natural real V33() V34() V35() FinSequence-like FinSequence-membered ext-real non negative complex-yielding V124() V125() V126() V133() V134() V135() V136() V137() V138() V139() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) holds
VFuncdiff (((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) (#) f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = - ((1 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * |[((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) / ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ^2) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:111
for f1, f2, f3, g1, g2, g3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff (((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = |[(((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + |[(((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:112
for f1, f2, f3, g1, g2, g3, h1, h2, h3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff ((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (|[((diff (h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((diff (h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + |[((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + |[((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:113
for f1, f2, f3, g1, g2, g3, h1, h2, h3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff ((((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - ((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - ((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,(((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - ((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (|[((((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + |[((((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + |[((((diff (h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (((diff (h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((diff (h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (((diff (h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,((((diff (h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - (((diff (h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;

theorem :: EUCLID_8:114
for f1, f2, f3, g1, g2, g3, h1, h2, h3 being ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,)
for t0 being ( ( ) ( real V33() ext-real ) Real) st f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) & h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) is_differentiable_in t0 : ( ( ) ( real V33() ext-real ) Real) holds
VFuncdiff (((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) - (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) (#) g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) Element of K19(K20(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ,REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( Relation-like complex-yielding V124() V125() ) set ) ) : ( ( ) ( ) set ) ) ,t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) = (|[(((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (diff (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + |[(((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((diff (f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) + |[(((diff (h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((diff (h3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ,(((diff (h1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f1 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((diff (h2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) ,t0 : ( ( ) ( real V33() ext-real ) Real) )) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (((f2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) - ((f3 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) * (g2 : ( ( V6() ) ( Relation-like V6() complex-yielding V124() V125() ) PartFunc of ,) . t0 : ( ( ) ( real V33() ext-real ) Real) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( real V33() ext-real ) Element of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V11() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) -valued V6() V38() V45(3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V124() V125() ) Element of REAL 3 : ( ( ) ( V11() ordinal natural real V33() V34() V35() ext-real non negative V133() V134() V135() V136() V137() V138() ) Element of NAT : ( ( ) ( V133() V134() V135() V136() V137() V138() V139() ) Element of K19(REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V9() V11() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V11() V38() V133() V134() V135() V139() ) set ) ) ) ;