:: TOPREAL9 semantic presentation

begin

theorem :: TOPREAL9:1
canceled;

theorem :: TOPREAL9:2
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for x, y, z being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) + z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TOPREAL9:3
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st not n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) is empty holds
x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) + (1.REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: TOPREAL9:4
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y, z being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) = ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) - r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (r : ( ( real ) ( V11() real ext-real ) number ) * z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
( ( not x : ( ( ) ( ) set ) = y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) or r : ( ( real ) ( V11() real ext-real ) number ) = 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) or y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) & ( ( r : ( ( real ) ( V11() real ext-real ) number ) = 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) or y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) implies x : ( ( ) ( ) set ) = y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) & ( not x : ( ( ) ( ) set ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) or r : ( ( real ) ( V11() real ext-real ) number ) = 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) or y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) & ( ( r : ( ( real ) ( V11() real ext-real ) number ) = 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) or y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) implies x : ( ( ) ( ) set ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TOPREAL9:5
for f being ( ( V16() V21() FinSequence-like real-valued ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) FinSequence) holds |.f : ( ( V16() V21() FinSequence-like real-valued ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) FinSequence) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ^2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) = Sum (sqr f : ( ( V16() V21() FinSequence-like real-valued ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) FinSequence) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ;

theorem :: TOPREAL9:6
for r being ( ( real ) ( V11() real ext-real ) number )
for M being ( ( non empty Reflexive discerning V103() triangle ) ( non empty Reflexive discerning V103() triangle ) MetrSpace)
for z1, z2, z3 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st z1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) <> z2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & z1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in cl_Ball (z3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty Reflexive discerning V103() triangle ) ( non empty Reflexive discerning V103() triangle ) MetrSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & z2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in cl_Ball (z3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty Reflexive discerning V103() triangle ) ( non empty Reflexive discerning V103() triangle ) MetrSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
r : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;

begin

definition
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real ) ( V11() real ext-real ) number ) ;
func Ball (x,r) -> ( ( ) ( ) Subset of ) equals :: TOPREAL9:def 1
 { p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) where p is ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ) RLTopStruct ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( ) set ) ) : |.(p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) < r : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( V16() V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ) : ( ( ) ( ) set ) ) } ;
func cl_Ball (x,r) -> ( ( ) ( ) Subset of ) equals :: TOPREAL9:def 2
 { p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) where p is ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ) RLTopStruct ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( ) set ) ) : |.(p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) <= r : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( V16() V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ) : ( ( ) ( ) set ) ) } ;
func Sphere (x,r) -> ( ( ) ( ) Subset of ) equals :: TOPREAL9:def 3
 { p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) where p is ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ) RLTopStruct ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( ) set ) ) : |.(p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( ) Element of n : ( ( ) ( ) RLTopStruct ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) = r : ( ( V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) ( V16() V21() V30(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) ) Element of K6(K7(K7(n : ( ( ) ( ) RLTopStruct ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ,n : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( V16() ) set ) ) : ( ( ) ( ) set ) ) } ;
end;

theorem :: TOPREAL9:7
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y, x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) iff |.(y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: TOPREAL9:8
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y, x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) iff |.(y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: TOPREAL9:9
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y, x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) iff |.(y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: TOPREAL9:10
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Ball ((0. (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V69( TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) holds
|.y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) ;

theorem :: TOPREAL9:11
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in cl_Ball ((0. (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V69( TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) holds
|.y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) ;

theorem :: TOPREAL9:12
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere ((0. (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V69( TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) holds
|.y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) ;

theorem :: TOPREAL9:13
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = e : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
Ball (e : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( bounded ) Element of K6( the carrier of (Euclid b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V103() triangle ) ( non empty strict Reflexive discerning V103() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) ;

theorem :: TOPREAL9:14
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = e : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
cl_Ball (e : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (Euclid b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V103() triangle ) ( non empty strict Reflexive discerning V103() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) ;

theorem :: TOPREAL9:15
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = e : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
Sphere (e : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (Euclid b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V103() triangle ) ( non empty strict Reflexive discerning V103() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) ;

theorem :: TOPREAL9:16
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) c= cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) ;

theorem :: TOPREAL9:17
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) c= cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) ;

theorem :: TOPREAL9:18
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds (Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Subset of ) \/ (Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) ;

theorem :: TOPREAL9:19
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) misses Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) ;

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real non positive ) ( V11() real ext-real non positive ) number ) ;
cluster Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real non positive ) ( V11() real ext-real non positive ) set ) ) : ( ( ) ( ) Subset of ) -> empty ;
end;

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real positive ) ( non empty V11() real ext-real positive non negative ) number ) ;
cluster Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real positive ) ( non empty V11() real ext-real positive non negative ) set ) ) : ( ( ) ( ) Subset of ) -> non empty ;
end;

theorem :: TOPREAL9:20
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st not Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) is empty holds
r : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPREAL9:21
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) is empty holds
r : ( ( real ) ( V11() real ext-real ) number ) <= 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real negative ) ( non empty V11() real ext-real non positive negative ) number ) ;
cluster cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real negative ) ( non empty V11() real ext-real non positive negative ) set ) ) : ( ( ) ( ) Subset of ) -> empty ;
end;

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real non negative ) ( V11() real ext-real non negative ) number ) ;
cluster cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real non negative ) ( V11() real ext-real non negative ) set ) ) : ( ( ) ( ) Subset of ) -> non empty ;
end;

theorem :: TOPREAL9:22
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st not cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) is empty holds
r : ( ( real ) ( V11() real ext-real ) number ) >= 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPREAL9:23
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) is empty holds
r : ( ( real ) ( V11() real ext-real ) number ) < 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPREAL9:24
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for x, z, y being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) + b : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) = 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) & (abs a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) + (abs b : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) = 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) & b : ( ( real ) ( V11() real ext-real ) number ) <> 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in cl_Ball (z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) & y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Ball (z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) holds
(a : ( ( real ) ( V11() real ext-real ) number ) * x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (b : ( ( real ) ( V11() real ext-real ) number ) * y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) ;

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real ) ( V11() real ext-real ) number ) ;
cluster Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( ) Subset of ) -> open ;
cluster cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( ) Subset of ) -> closed ;
cluster Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( ) Subset of ) -> closed ;
end;

registration
let n be ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real ) ( V11() real ext-real ) number ) ;
cluster Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( open ) Subset of ) -> bounded ;
cluster cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( closed ) Subset of ) -> bounded ;
cluster Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( closed ) Subset of ) -> bounded ;
end;

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real ) ( V11() real ext-real ) number ) ;
cluster Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( open ) Subset of ) -> convex ;
cluster cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( closed ) Subset of ) -> convex ;
end;

definition
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let f be ( ( V21() V30( the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ( V16() V21() V30( the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
attr f is homogeneous means :: TOPREAL9:def 4
 for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( non empty ) ( non empty ) MetrStruct ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( ) set ) ) holds f : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) . (r : ( ( real ) ( V11() real ext-real ) number ) * x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( non empty ) ( non empty ) MetrStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( non empty ) ( non empty ) MetrStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) * (f : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) . x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( non empty ) ( non empty ) MetrStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( non empty ) ( non empty ) MetrStruct ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ;
end;

registration
let n be ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;
cluster (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) --> (0. (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V69( TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( V21() V30( the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ( V16() V21() V30( the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) continuous ) Element of K6(K7( the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() ) set ) ) : ( ( ) ( ) set ) ) -> V21() V30( the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) additive homogeneous ;
end;

registration
let n be ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;
cluster V16() V21() V30( the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) continuous additive homogeneous for ( ( ) ( ) Element of K6(K7( the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let a, c be ( ( real ) ( V11() real ext-real ) number ) ;
cluster AffineMap (a : ( ( real ) ( V11() real ext-real ) set ) ,0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ,c : ( ( real ) ( V11() real ext-real ) set ) ,0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ( V16() V21() V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() ) set ) ) : ( ( ) ( ) set ) ) -> V21() V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) additive homogeneous ;
end;

theorem :: TOPREAL9:25
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( V21() V30( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V21() V30( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X being ( ( convex ) ( convex ) Subset of ) holds f : ( ( V21() V30( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V21() V30( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) .: X : ( ( convex ) ( convex ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is convex ;

definition
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let p, q be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
func halfline (p,q) -> ( ( ) ( ) Subset of ) equals :: TOPREAL9:def 5
 { (((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) - l : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * p : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (l : ( ( ) ( V11() real ext-real ) Real) * q : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) where l is ( ( ) ( V11() real ext-real ) Real) : 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) <= l : ( ( ) ( V11() real ext-real ) Real) } ;
end;

theorem :: TOPREAL9:26
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for p, q being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Subset of ) iff ex l being ( ( real ) ( V11() real ext-real ) number ) st
( x : ( ( ) ( ) set ) = ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) - l : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (l : ( ( real ) ( V11() real ext-real ) number ) * q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) <= l : ( ( real ) ( V11() real ext-real ) number ) ) ) ;

registration
let n be ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let p, q be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
cluster halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Subset of ) -> non empty ;
end;

theorem :: TOPREAL9:27
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for p, q being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) ;

theorem :: TOPREAL9:28
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for q, p being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) ;

theorem :: TOPREAL9:29
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for p being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) = {p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) ;

theorem :: TOPREAL9:30
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for x, p, q being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) holds
halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) c= halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) ;

theorem :: TOPREAL9:31
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for x, p, q being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) & x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) = halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) ;

theorem :: TOPREAL9:32
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for p, q being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds LSeg (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) ;

registration
let n be ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let p, q be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
cluster halfline (p : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Subset of ) -> convex ;
end;

theorem :: TOPREAL9:33
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y, x, z being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) & z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( open bounded convex ) Subset of ) holds
(LSeg (y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( closed bounded ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) ;

theorem :: TOPREAL9:34
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y, x, z being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) & z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) holds
(LSeg (y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \ {y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( open bounded convex ) Subset of ) ;

theorem :: TOPREAL9:35
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y, x, z being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) & z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) holds
(LSeg (y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( closed bounded ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) ;

theorem :: TOPREAL9:36
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for y, x, z being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) & z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) holds
(halfline (y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty convex ) Subset of ) /\ (Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( closed bounded ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) ;

theorem :: TOPREAL9:37
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r, a being ( ( real ) ( V11() real ext-real ) number )
for y, z, x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for S, T, X being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) st S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & X : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( open bounded convex ) Subset of ) & a : ( ( real ) ( V11() real ext-real ) number ) = ((- (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * |((z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) + (sqrt (delta ((Sum (sqr (T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ,(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * |((z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ,((Sum (sqr (S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - X : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) - (r : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * (Sum (sqr (T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) holds
ex e being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st
( {e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) = (halfline (y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty convex ) Subset of ) /\ (Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( closed bounded ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (a : ( ( real ) ( V11() real ext-real ) number ) * z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPREAL9:38
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r, a being ( ( real ) ( V11() real ext-real ) number )
for y, z, x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for S, T, Y being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) st S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & Y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) & z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in cl_Ball (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded convex ) Subset of ) holds
ex e being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st
( e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & {y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) = (halfline (y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty convex ) Subset of ) /\ (Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( closed bounded ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) implies e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) & ( not z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) & a : ( ( real ) ( V11() real ext-real ) number ) = ((- (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * |((z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - (((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) - x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) + (sqrt (delta ((Sum (sqr (T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ,(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * |((z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - (((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) - x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ,((Sum (sqr (S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - Y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) - (r : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * (Sum (sqr (T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) implies e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * (((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (a : ( ( real ) ( V11() real ext-real ) number ) * z : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ) ;

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real negative ) ( non empty V11() real ext-real non positive negative ) number ) ;
cluster Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real negative ) ( non empty V11() real ext-real non positive negative ) set ) ) : ( ( ) ( closed ) Subset of ) -> empty ;
end;

registration
let n be ( ( non empty natural ) ( non empty ordinal natural V11() real ext-real positive non negative ) Nat) ;
let x be ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( non empty natural ) ( non empty ordinal natural V11() real ext-real positive non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ;
let r be ( ( real non negative ) ( V11() real ext-real non negative ) number ) ;
cluster Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(n : ( ( non empty natural ) ( non empty ordinal natural V11() real ext-real positive non negative ) set ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL n : ( ( non empty natural ) ( non empty ordinal natural V11() real ext-real positive non negative ) set ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real non negative ) ( V11() real ext-real non negative ) set ) ) : ( ( ) ( closed ) Subset of ) -> non empty ;
end;

theorem :: TOPREAL9:39
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st not Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) is empty holds
r : ( ( real ) ( V11() real ext-real ) number ) >= 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPREAL9:40
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st not n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) is empty & Sphere (x : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) is empty holds
r : ( ( real ) ( V11() real ext-real ) number ) < 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;

begin

theorem :: TOPREAL9:41
for a, b being ( ( real ) ( V11() real ext-real ) number )
for s, t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds ((a : ( ( real ) ( V11() real ext-real ) number ) * s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (b : ( ( real ) ( V11() real ext-real ) number ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) = (a : ( ( real ) ( V11() real ext-real ) number ) * (s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) + (b : ( ( real ) ( V11() real ext-real ) number ) * (t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ;

theorem :: TOPREAL9:42
for a, b being ( ( real ) ( V11() real ext-real ) number )
for s, t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds ((a : ( ( real ) ( V11() real ext-real ) number ) * s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (b : ( ( real ) ( V11() real ext-real ) number ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) = (a : ( ( real ) ( V11() real ext-real ) number ) * (s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) + (b : ( ( real ) ( V11() real ext-real ) number ) * (t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ;

theorem :: TOPREAL9:43
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) iff |.(t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: TOPREAL9:44
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) iff |.(t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: TOPREAL9:45
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) holds
( t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) iff |.(t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) ) ;

registration
let a, b be ( ( real ) ( V11() real ext-real ) number ) ;
let r be ( ( real positive ) ( non empty V11() real ext-real positive non negative ) number ) ;
cluster inside_of_circle (a : ( ( real ) ( V11() real ext-real ) set ) ,b : ( ( real ) ( V11() real ext-real ) set ) ,r : ( ( real positive ) ( non empty V11() real ext-real positive non negative ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -> non empty ;
end;

registration
let a, b be ( ( real ) ( V11() real ext-real ) number ) ;
let r be ( ( real non negative ) ( V11() real ext-real non negative ) number ) ;
cluster closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) set ) ,b : ( ( real ) ( V11() real ext-real ) set ) ,r : ( ( real non negative ) ( V11() real ext-real non negative ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -> non empty ;
end;

theorem :: TOPREAL9:46
for a, b, r being ( ( real ) ( V11() real ext-real ) number ) holds circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:47
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
cl_Ball (x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V103() triangle ) ( non empty strict Reflexive discerning V103() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:48
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
Ball (x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( bounded ) Element of K6( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V103() triangle ) ( non empty strict Reflexive discerning V103() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:49
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
Sphere (x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V103() triangle ) ( non empty strict Reflexive discerning V103() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:50
for a, b, r being ( ( real ) ( V11() real ext-real ) number ) holds Ball (|[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( open bounded convex ) Subset of ) = inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:51
for a, b, r being ( ( real ) ( V11() real ext-real ) number ) holds cl_Ball (|[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded convex ) Subset of ) = closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:52
for a, b, r being ( ( real ) ( V11() real ext-real ) number ) holds Sphere (|[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) = circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:53
for a, b, r being ( ( real ) ( V11() real ext-real ) number ) holds inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:54
for a, b, r being ( ( real ) ( V11() real ext-real ) number ) holds inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) misses circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:55
for a, b, r being ( ( real ) ( V11() real ext-real ) number ) holds (inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:56
for r being ( ( real ) ( V11() real ext-real ) number )
for s being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere ((0. (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V69( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) ) complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) holds
((s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) + ((s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) ^2 : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: TOPREAL9:57
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for s, t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
r : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V16() non-zero V18() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( RAT : ( ( ) ( V179() V180() V181() V182() V185() ) set ) ) V21() V22() constant V24() V33() V34() V37() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V179() V180() V181() V182() V183() V184() V185() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPREAL9:58
for a, b, w, r being ( ( real ) ( V11() real ext-real ) number )
for s, t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for S, T, X being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) st S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & X : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & w : ( ( real ) ( V11() real ext-real ) number ) = ((- (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * |((t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) + (sqrt (delta ((Sum (sqr (T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ,(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * |((t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ,((Sum (sqr (S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - X : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) - (r : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * (Sum (sqr (T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) & s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex e being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st
( {e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) = (halfline (s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty convex ) Subset of ) /\ (circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) - w : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (w : ( ( real ) ( V11() real ext-real ) number ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPREAL9:59
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for s, t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(LSeg (s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) ;

theorem :: TOPREAL9:60
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for s, t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(LSeg (s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \ {s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL9:61
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for s, t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(LSeg (s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) ;

theorem :: TOPREAL9:62
for a, b, r being ( ( real ) ( V11() real ext-real ) number )
for s, t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(halfline (s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty convex ) Subset of ) /\ (circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) ;

theorem :: TOPREAL9:63
for a, b, r, w being ( ( real ) ( V11() real ext-real ) number )
for s, t being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) )
for S, T, Y being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) st S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & Y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) = |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex e being ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) st
( e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) <> s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) & {s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty V24() ) set ) = (halfline (s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty convex ) Subset of ) /\ (circle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,r : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (|[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) implies e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) & ( not t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) in Sphere (|[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( closed bounded ) Subset of ) & w : ( ( real ) ( V11() real ext-real ) number ) = ((- (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * |((t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - (((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) - |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) + (sqrt (delta ((Sum (sqr (T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ,(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * |((t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) - (((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) - |[a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ,((Sum (sqr (S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - Y : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) - (r : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) * (Sum (sqr (T : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) - S : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) V21() V37() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V179() V180() V181() V185() ) set ) : ( ( ) ( V24() FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) implies e : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) = ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) - w : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * (((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * s : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + ((1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) + (w : ( ( real ) ( V11() real ext-real ) number ) * t : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) V21() V37() V44(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ) ;

registration
let a, b, r be ( ( real ) ( V11() real ext-real ) number ) ;
cluster inside_of_circle (a : ( ( real ) ( V11() real ext-real ) set ) ,b : ( ( real ) ( V11() real ext-real ) set ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -> convex ;
cluster closed_inside_of_circle (a : ( ( real ) ( V11() real ext-real ) set ) ,b : ( ( real ) ( V11() real ext-real ) set ) ,r : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V33() V34() V179() V180() V181() V182() V183() V184() ) Element of NAT : ( ( ) ( V179() V180() V181() V182() V183() V184() V185() ) Element of K6(REAL : ( ( ) ( V179() V180() V181() V185() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V191() V192() V209() V210() V211() V212() V213() V214() V215() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -> convex ;
end;