:: ANPROJ_2 semantic presentation

begin

theorem :: ANPROJ_2:1
for V being ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for u, v, w being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for a, b, c being ( ( ) ( V24() V25() ) Real) st ((a : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) holds
( not u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & not v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & not w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & not u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & not are_Prop u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: ANPROJ_2:2
for V being ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for u, v, u1, v1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for a, b, a1, b1 being ( ( ) ( V24() V25() ) Real) st (((a : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (a1 : ( ( ) ( V24() V25() ) Real) * u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b1 : ( ( ) ( V24() V25() ) Real) * v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & a1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) holds
( not u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & not v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & not are_Prop u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & not v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & not are_Prop u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & not u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep ) ;

theorem :: ANPROJ_2:3
for V being ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for p, q, r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for w being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex a, b, c being ( ( ) ( V24() V25() ) Real) st w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = ((a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) & ( for a, b, c being ( ( ) ( V24() V25() ) Real) st ((a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) holds
for u, u1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex y being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & not y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero ) ;

theorem :: ANPROJ_2:4
for V being ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for p, q, r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for w being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex a, b, c, d being ( ( ) ( V24() V25() ) Real) st w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = (((a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (d : ( ( ) ( V24() V25() ) Real) * s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) & ( for a, b, c, d being ( ( ) ( V24() V25() ) Real) st (((a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) holds
for u, v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & not v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero holds
ex y, w being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & not y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & not w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero ) ;

theorem :: ANPROJ_2:5
for V being ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for u, v, u1, v1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for a, b, a1, b1 being ( ( ) ( V24() V25() ) Real) st (((a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (a1 : ( ( ) ( V24() V25() ) Real) * u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b1 : ( ( ) ( V24() V25() ) Real) * v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & a1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) holds
for y being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero or not u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep or not u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep ) ;

definition
let V be ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
let u, v, w be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
pred u,v,w are_Prop_Vect means :: ANPROJ_2:def 1
( not u : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) is zero & not v : ( ( Function-like quasi_total ) ( V7() V10(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) is zero & not w : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) is zero );
end;

definition
let V be ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
let u, v, w, u1, v1, w1 be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
pred u,v,w,u1,v1,w1 lie_on_a_triangle means :: ANPROJ_2:def 2
( u : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,v : ( ( Function-like quasi_total ) ( V7() V10(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,w1 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep & u : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,w : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep & v : ( ( Function-like quasi_total ) ( V7() V10(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,w : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,u1 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep );
end;

definition
let V be ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
let o, u, v, w, u2, v2, w2 be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
pred o,u,v,w,u2,v2,w2 are_perspective means :: ANPROJ_2:def 3
( o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,u : ( ( Function-like quasi_total ) ( V7() V10(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep & o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,v : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep & o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,w : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) ,w2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep );
end;

theorem :: ANPROJ_2:6
for V being ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for o, u, u2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & not are_Prop o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not are_Prop o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not are_Prop u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_Prop_Vect holds
( ex a1, b1 being ( ( ) ( V24() V25() ) Real) st
( b1 : ( ( ) ( V24() V25() ) Real) * u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) + (a1 : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) & a1 : ( ( ) ( V24() V25() ) Real) <> 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b1 : ( ( ) ( V24() V25() ) Real) <> 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & ex a2, c2 being ( ( ) ( V24() V25() ) Real) st
( u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = (c2 : ( ( ) ( V24() V25() ) Real) * o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (a2 : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) & c2 : ( ( ) ( V24() V25() ) Real) <> 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & a2 : ( ( ) ( V24() V25() ) Real) <> 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: ANPROJ_2:7
for V being ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for p, q, r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & not are_Prop p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_Prop_Vect holds
ex a, b being ( ( ) ( V24() V25() ) Real) st r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = (a : ( ( ) ( V24() V25() ) Real) * p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: ANPROJ_2:8
for V being ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for o, u, v, w, u2, v2, w2, u1, v1, w1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_Prop_Vect & u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_Prop_Vect & u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_Prop_Vect & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_perspective & not are_Prop o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not are_Prop o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not are_Prop o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not are_Prop u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not are_Prop v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not are_Prop w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & not o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & not o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) lie_on_a_triangle & u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) lie_on_a_triangle holds
u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep ;

definition
let V be ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
let o, u, v, w, u2, v2, w2 be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
pred o,u,v,w,u2,v2,w2 lie_on_an_angle means :: ANPROJ_2:def 4
( not o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,u : ( ( Function-like quasi_total ) ( V7() V10(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep & o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,u : ( ( Function-like quasi_total ) ( V7() V10(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,v : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) are_LinDep & o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,u : ( ( Function-like quasi_total ) ( V7() V10(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep & o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,u2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) ,v2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep & o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,u2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) ,w2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) are_LinDep );
end;

definition
let V be ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
let o, u, v, w, u2, v2, w2 be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
pred o,u,v,w,u2,v2,w2 are_half_mutually_not_Prop means :: ANPROJ_2:def 5
( not are_Prop o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,v : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) & not are_Prop o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,w : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) & not are_Prop o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,v2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) & not are_Prop o : ( ( ) ( ) Relation3 of V : ( ( ) ( ) CollStr ) ) ,w2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) & not are_Prop u : ( ( Function-like quasi_total ) ( V7() V10(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,v : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) & not are_Prop u : ( ( Function-like quasi_total ) ( V7() V10(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(V : ( ( ) ( ) CollStr ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) & not are_Prop u2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) ,v2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) & not are_Prop u2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) ,w2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) & not are_Prop v : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) V11(V : ( ( ) ( ) CollStr ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) CollStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) & not are_Prop v2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) ,w2 : ( ( ) ( ) Element of V : ( ( ) ( ) CollStr ) ) );
end;

theorem :: ANPROJ_2:9
for V being ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for o, u, v, w, u2, v2, w2, u1, v1, w1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is zero & u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_Prop_Vect & u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_Prop_Vect & u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_Prop_Vect & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) lie_on_an_angle & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_half_mutually_not_Prop & u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep & w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep holds
u1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) are_LinDep ;

theorem :: ANPROJ_2:10
for A being ( ( non empty ) ( non empty ) set )
for x1 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) ex f being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) st
( f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( ) set ) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( ) set ) <> x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( ) set ) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: ANPROJ_2:11
for A being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) )
for x1, x2, x3 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V24() V25() ) Real) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V24() V25() ) Real) <> x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V24() V25() ) Real) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V24() V25() ) Real) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V24() V25() ) Real) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V24() V25() ) Real) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V24() V25() ) Real) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V24() V25() ) Real) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V24() V25() ) Real) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) holds
for a, b, c being ( ( ) ( V24() V25() ) Real) st (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = RealFuncZero A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: ANPROJ_2:12
for A being ( ( non empty ) ( non empty ) set )
for x1, x2, x3 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
ex f, g, h being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) st
for a, b, c being ( ( ) ( V24() V25() ) Real) st (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = RealFuncZero A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: ANPROJ_2:13
for A being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) )
for x1, x2, x3 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st A : ( ( non empty ) ( non empty ) set ) = {x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of K32(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) <> x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) holds
for h9 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ex a, b, c being ( ( ) ( V24() V25() ) Real) st h9 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ;

theorem :: ANPROJ_2:14
for A being ( ( non empty ) ( non empty ) set )
for x1, x2, x3 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st A : ( ( non empty ) ( non empty ) set ) = {x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of K32(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
ex f, g, h being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) st
for h9 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ex a, b, c being ( ( ) ( V24() V25() ) Real) st h9 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ;

theorem :: ANPROJ_2:15
for A being ( ( non empty ) ( non empty ) set )
for x1, x2, x3 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st A : ( ( non empty ) ( non empty ) set ) = {x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of K32(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
ex f, g, h being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) st
( ( for a, b, c being ( ( ) ( V24() V25() ) Real) st (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = RealFuncZero A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) holds
( a : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) & ( for h9 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ex a, b, c being ( ( ) ( V24() V25() ) Real) st h9 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) ) ;

theorem :: ANPROJ_2:16
ex V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ex u, v, w being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
( ( for a, b, c being ( ( ) ( V24() V25() ) Real) st ((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) = 0. V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) & ( for y being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ex a, b, c being ( ( ) ( V24() V25() ) Real) st y : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) = ((a : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: ANPROJ_2:17
for A being ( ( non empty ) ( non empty ) set )
for f, g, h, f1 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) )
for x1, x2, x3, x4 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V24() V25() ) Real) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V24() V25() ) Real) <> x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V24() V25() ) Real) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V24() V25() ) Real) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V24() V25() ) Real) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V24() V25() ) Real) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V24() V25() ) Real) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V24() V25() ) Real) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V24() V25() ) Real) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V24() V25() ) Real) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V24() V25() ) Real) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V24() V25() ) Real) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) holds
for a, b, c, d being ( ( ) ( V24() V25() ) Real) st (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [d : ( ( ) ( V24() V25() ) Real) ,f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = RealFuncZero A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & d : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: ANPROJ_2:18
for A being ( ( non empty ) ( non empty ) set )
for x1, x2, x3, x4 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
ex f, g, h, f1 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) st
for a, b, c, d being ( ( ) ( V24() V25() ) Real) st (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [d : ( ( ) ( V24() V25() ) Real) ,f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = RealFuncZero A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & d : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: ANPROJ_2:19
for A being ( ( non empty ) ( non empty ) set )
for f, g, h, f1 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) )
for x1, x2, x3, x4 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st A : ( ( non empty ) ( non empty ) set ) = {x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of K32(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) <> x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() V25() ) Element of REAL : ( ( ) ( non empty V35() ) set ) ) = 1 : ( ( ) ( non empty V17() V18() V19() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ( for z being ( ( ) ( ) set ) st z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) in A : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) . z : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) holds
for h9 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ex a, b, c, d being ( ( ) ( V24() V25() ) Real) st h9 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [d : ( ( ) ( V24() V25() ) Real) ,f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ;

theorem :: ANPROJ_2:20
for A being ( ( non empty ) ( non empty ) set )
for x1, x2, x3, x4 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st A : ( ( non empty ) ( non empty ) set ) = {x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of K32(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
ex f, g, h, f1 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) st
for h9 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ex a, b, c, d being ( ( ) ( V24() V25() ) Real) st h9 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [d : ( ( ) ( V24() V25() ) Real) ,f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ;

theorem :: ANPROJ_2:21
for A being ( ( non empty ) ( non empty ) set )
for x1, x2, x3, x4 being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st A : ( ( non empty ) ( non empty ) set ) = {x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of K32(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & x3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> x4 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
ex f, g, h, f1 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) st
( ( for a, b, c, d being ( ( ) ( V24() V25() ) Real) st (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [d : ( ( ) ( V24() V25() ) Real) ,f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = RealFuncZero A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) holds
( a : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & d : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) & ( for h9 being ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ex a, b, c, d being ( ( ) ( V24() V25() ) Real) st h9 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) = (RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncAdd A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . (((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( V24() V25() ) Real) ,f : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [b : ( ( ) ( V24() V25() ) Real) ,g : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [c : ( ( ) ( V24() V25() ) Real) ,h : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) )) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ,((RealFuncExtMult A : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V7() V10(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) V11( Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) Function-like quasi_total ) Element of K32(K33(K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) . [d : ( ( ) ( V24() V25() ) Real) ,f1 : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ] : ( ( ) ( ) Element of K33(REAL : ( ( ) ( non empty V35() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) : ( ( ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() ) set ) ) Function-like quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() ) set ) ) ) ) ) ;

theorem :: ANPROJ_2:22
ex V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ex u, v, w, u1 being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
( ( for a, b, c, d being ( ( ) ( V24() V25() ) Real) st (((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (d : ( ( ) ( V24() V25() ) Real) * u1 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) = 0. V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & d : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) & ( for y being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ex a, b, c, d being ( ( ) ( V24() V25() ) Real) st y : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) = (((a : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (d : ( ( ) ( V24() V25() ) Real) * u1 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

definition
let IT be ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
attr IT is up-3-dimensional means :: ANPROJ_2:def 6
 ex u, v, w1 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st
for a, b, c being ( ( ) ( V24() V25() ) Real) st ((a : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of IT : ( ( ) ( ) CollStr ) : ( ( ) ( ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of IT : ( ( ) ( ) CollStr ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the U1 of IT : ( ( ) ( ) CollStr ) : ( ( ) ( ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of IT : ( ( ) ( ) CollStr ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the U1 of IT : ( ( ) ( ) CollStr ) : ( ( ) ( ) set ) ) = 0. IT : ( ( ) ( ) CollStr ) : ( ( ) ( ) Element of the U1 of IT : ( ( ) ( ) CollStr ) : ( ( ) ( ) set ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) );
end;

registration
cluster non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional for ( ( ) ( ) RLSStruct ) ;
end;

registration
cluster non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional -> non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital for ( ( ) ( ) RLSStruct ) ;
end;

definition
let CS be ( ( non empty ) ( non empty ) CollStr ) ;
redefine attr CS is reflexive means :: ANPROJ_2:def 7
 for p, q, r being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear );
redefine attr CS is transitive means :: ANPROJ_2:def 8
 for p, q, r, r1, r2 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear holds
r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ;
end;

definition
let IT be ( ( non empty ) ( non empty ) CollStr ) ;
attr IT is Vebleian means :: ANPROJ_2:def 9
 for p, p1, p2, r, r1 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear holds
ex r2 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear );
attr IT is at_least_3rank means :: ANPROJ_2:def 10
 for p, q being ( ( ) ( ) Element of ( ( ) ( ) set ) ) ex r being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear );
end;

theorem :: ANPROJ_2:23
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for p, q, r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear iff ex u, v, w being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = Dir u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of K32(K175(b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ) : ( ( ) ( non empty ) Element of K32( the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = Dir v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of K32(K175(b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ) : ( ( ) ( non empty ) Element of K32( the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) & r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = Dir w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of K32(K175(b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ) : ( ( ) ( non empty ) Element of K32( the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) & not u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) is zero & not v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) is zero & not w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) is zero & u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ,w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) are_LinDep ) ) ;

registration
let V be ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
cluster ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( strict ) ( non empty strict ) CollStr ) -> strict reflexive transitive ;
end;

theorem :: ANPROJ_2:24
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for p, q, r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ) ;

registration
let V be ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
cluster ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( strict ) ( non empty strict reflexive transitive ) CollStr ) -> strict Vebleian ;
end;

registration
let V be ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) RealLinearSpace) ;
cluster ProjectiveSpace V : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) RLSStruct ) : ( ( strict ) ( non empty strict reflexive transitive Vebleian ) CollStr ) -> strict proper ;
end;

theorem :: ANPROJ_2:25
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) st ex u, v being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
for a, b being ( ( ) ( V24() V25() ) Real) st (a : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) = 0. V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) holds
ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( strict ) ( non empty strict reflexive transitive Vebleian ) CollStr ) is at_least_3rank ;

registration
let V be ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) RealLinearSpace) ;
cluster ProjectiveSpace V : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) RLSStruct ) : ( ( strict ) ( non empty strict reflexive transitive proper Vebleian ) CollStr ) -> strict at_least_3rank ;
end;

registration
cluster non empty reflexive transitive proper Vebleian at_least_3rank for ( ( ) ( ) CollStr ) ;
end;

definition
mode CollProjectiveSpace is ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollStr ) ;
end;

definition
let IT be ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ;
attr IT is Fanoian means :: ANPROJ_2:def 11
 for p1, r2, q, r1, q1, p, r being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & not p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & not p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & not p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear holds
r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ;
attr IT is Desarguesian means :: ANPROJ_2:def 12
 for o, p1, p2, p3, q1, q2, q3, r1, r2, r3 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & not o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & not o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear holds
r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ;
attr IT is Pappian means :: ANPROJ_2:def 13
 for o, p1, p2, p3, q1, q2, q3, r1, r2, r3 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & o : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear holds
r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ;
end;

definition
let IT be ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ;
attr IT is 2-dimensional means :: ANPROJ_2:def 14
 for p, p1, q, q1 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) ex r being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear );
end;

notation
let IT be ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ;
antonym up-3-dimensional IT for 2-dimensional ;
end;

definition
let IT be ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ;
attr IT is at_most-3-dimensional means :: ANPROJ_2:def 15
 for p, p1, q, q1, r2 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) ex r, r1 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear );
end;

theorem :: ANPROJ_2:26
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for p1, r2, q, r1, q1, p, r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & not p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & not p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & not p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear holds
r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ;

registration
let V be ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) RealLinearSpace) ;
cluster ProjectiveSpace V : ( ( non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional ) RLSStruct ) : ( ( strict ) ( non empty strict reflexive transitive proper Vebleian at_least_3rank ) CollStr ) -> strict Fanoian Desarguesian Pappian ;
end;

theorem :: ANPROJ_2:27
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) st ex u, v, w being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
( ( for a, b, c being ( ( ) ( V24() V25() ) Real) st ((a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) = 0. V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) & ( for y being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ex a, b, c being ( ( ) ( V24() V25() ) Real) st y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = ((a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) ) holds
ex x1, x2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & ( for r1, r2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ) ) ) ;

theorem :: ANPROJ_2:28
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) st ex x1, x2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> x2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & ( for r1, r2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ) ) ) holds
for p, p1, q, q1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ) ;

theorem :: ANPROJ_2:29
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) st ex u, v, w being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
( ( for a, b, c being ( ( ) ( V24() V25() ) Real) st ((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * u : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) = 0. V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & c : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) & ( for y being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ex a, b, c being ( ( ) ( V24() V25() ) Real) st y : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) = ((a : ( ( ) ( V24() V25() ) Real) * u : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) ) holds
ex CS being ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) st
( CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) = ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( strict ) ( non empty strict reflexive transitive Vebleian ) CollStr ) & CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) is 2-dimensional ) ;

theorem :: ANPROJ_2:30
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) st ex y, u, v, w being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
( ( for w1 being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ex a, b, a1, b1 being ( ( ) ( V24() V25() ) Real) st w1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = (((a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (a1 : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b1 : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) & ( for a, b, a1, b1 being ( ( ) ( V24() V25() ) Real) st (((a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (a1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b1 : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) = 0. V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & a1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) ) holds
ex p, q1, q2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( not p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & ( for r1, r2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex q3, r3 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ) ) ) ;

theorem :: ANPROJ_2:31
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) st ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( strict ) ( non empty strict reflexive transitive Vebleian ) CollStr ) is proper & ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( strict ) ( non empty strict reflexive transitive Vebleian ) CollStr ) is at_least_3rank & ex p, q1, q2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( not p : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & ( for r1, r2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex q3, r3 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & p : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ,r3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ) ) ) holds
ex CS being ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) st
( CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) = ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( strict ) ( non empty strict reflexive transitive Vebleian ) CollStr ) & CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) is at_most-3-dimensional ) ;

theorem :: ANPROJ_2:32
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) st ex y, u, v, w being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
( ( for w1 being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ex a, b, c, c1 being ( ( ) ( V24() V25() ) Real) st w1 : ( ( ) ( V24() V25() ) Real) = (((a : ( ( ) ( V24() V25() ) Real) * y : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (c1 : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) & ( for a, b, a1, b1 being ( ( ) ( V24() V25() ) Real) st (((a : ( ( ) ( V24() V25() ) Real) * y : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * u : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (a1 : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b1 : ( ( ) ( V24() V25() ) Real) * w : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) = 0. V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & a1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) ) holds
ex CS being ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) st
( CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) = ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( strict ) ( non empty strict reflexive transitive Vebleian ) CollStr ) & CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) is at_most-3-dimensional ) ;

theorem :: ANPROJ_2:33
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) st ex u, v, u1, v1 being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
for a, b, a1, b1 being ( ( ) ( V24() V25() ) Real) st (((a : ( ( ) ( V24() V25() ) Real) * u : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (a1 : ( ( ) ( V24() V25() ) Real) * u1 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b1 : ( ( ) ( V24() V25() ) Real) * v1 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) = 0. V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & a1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) holds
ex CS being ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) st
( CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) = ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( strict ) ( non empty strict reflexive transitive Vebleian ) CollStr ) & not CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) is 2-dimensional ) ;

theorem :: ANPROJ_2:34
for V being ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) st ex u, v, u1, v1 being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st
( ( for w being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ex a, b, a1, b1 being ( ( ) ( V24() V25() ) Real) st w : ( ( ) ( V24() V25() ) Real) = (((a : ( ( ) ( V24() V25() ) Real) * u : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (a1 : ( ( ) ( V24() V25() ) Real) * u1 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b1 : ( ( ) ( V24() V25() ) Real) * v1 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) & ( for a, b, a1, b1 being ( ( ) ( V24() V25() ) Real) st (((a : ( ( ) ( V24() V25() ) Real) * u : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( V24() V25() ) Real) * v : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (a1 : ( ( ) ( V24() V25() ) Real) * u1 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) + (b1 : ( ( ) ( V24() V25() ) Real) * v1 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) = 0. V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( zero ) Element of the U1 of b1 : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty non trivial ) set ) ) holds
( a : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & a1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & b1 : ( ( ) ( V24() V25() ) Real) = 0 : ( ( ) ( empty V17() V18() V19() V21() V22() V23() V24() V25() ) Element of NAT : ( ( ) ( non empty V17() V18() V19() ) Element of K32(REAL : ( ( ) ( non empty V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ) ) holds
ex CS being ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) st
( CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) = ProjectiveSpace V : ( ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( strict ) ( non empty strict reflexive transitive Vebleian ) CollStr ) & CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) is up-3-dimensional & CS : ( ( non empty reflexive transitive proper Vebleian at_least_3rank ) ( non empty reflexive transitive proper Vebleian at_least_3rank ) CollProjectiveSpace) is at_most-3-dimensional ) ;

registration
cluster non empty strict reflexive transitive proper Vebleian at_least_3rank Fanoian Desarguesian Pappian 2-dimensional for ( ( ) ( ) CollStr ) ;
cluster non empty strict reflexive transitive proper Vebleian at_least_3rank Fanoian Desarguesian Pappian up-3-dimensional at_most-3-dimensional for ( ( ) ( ) CollStr ) ;
end;

definition
mode CollProjectivePlane is ( ( non empty reflexive transitive proper Vebleian at_least_3rank 2-dimensional ) ( non empty reflexive transitive proper Vebleian at_least_3rank 2-dimensional ) CollProjectiveSpace) ;
end;

theorem :: ANPROJ_2:35
for CS being ( ( non empty ) ( non empty ) CollStr ) holds
( CS : ( ( non empty ) ( non empty ) CollStr ) is ( ( non empty reflexive transitive proper Vebleian at_least_3rank 2-dimensional ) ( non empty reflexive transitive proper Vebleian at_least_3rank 2-dimensional ) CollProjectiveSpace) iff ( CS : ( ( non empty ) ( non empty ) CollStr ) is ( ( non empty reflexive transitive proper at_least_3rank ) ( non empty reflexive transitive proper at_least_3rank ) CollSp) & ( for p, p1, q, q1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_collinear ) ) ) ) ;