let X be non empty set ; :: thesis: for a being Complex

for F, G being VECTOR of (C_Algebra_of_BoundedFunctions X)

for f, g being Function of X,COMPLEX st f = F & g = G holds

( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

let a be Complex; :: thesis: for F, G being VECTOR of (C_Algebra_of_BoundedFunctions X)

for f, g being Function of X,COMPLEX st f = F & g = G holds

( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

let F, G be VECTOR of (C_Algebra_of_BoundedFunctions X); :: thesis: for f, g being Function of X,COMPLEX st f = F & g = G holds

( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

let f, g be Function of X,COMPLEX; :: thesis: ( f = F & g = G implies ( G = a * F iff for x being Element of X holds g . x = a * (f . x) ) )

assume A1: ( f = F & g = G ) ; :: thesis: ( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

A2: C_Algebra_of_BoundedFunctions X is ComplexSubAlgebra of CAlgebra X by Th2;

reconsider f1 = F, g1 = G as VECTOR of (CAlgebra X) by TARSKI:def 3;

then g1 = a * f1 by A1, CFUNCDOM:4;

hence G = a * F by A2, Th3; :: thesis: verum

for F, G being VECTOR of (C_Algebra_of_BoundedFunctions X)

for f, g being Function of X,COMPLEX st f = F & g = G holds

( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

let a be Complex; :: thesis: for F, G being VECTOR of (C_Algebra_of_BoundedFunctions X)

for f, g being Function of X,COMPLEX st f = F & g = G holds

( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

let F, G be VECTOR of (C_Algebra_of_BoundedFunctions X); :: thesis: for f, g being Function of X,COMPLEX st f = F & g = G holds

( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

let f, g be Function of X,COMPLEX; :: thesis: ( f = F & g = G implies ( G = a * F iff for x being Element of X holds g . x = a * (f . x) ) )

assume A1: ( f = F & g = G ) ; :: thesis: ( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

A2: C_Algebra_of_BoundedFunctions X is ComplexSubAlgebra of CAlgebra X by Th2;

reconsider f1 = F, g1 = G as VECTOR of (CAlgebra X) by TARSKI:def 3;

hereby :: thesis: ( ( for x being Element of X holds g . x = a * (f . x) ) implies G = a * F )

assume
for x being Element of X holds g . x = a * (f . x)
; :: thesis: G = a * Fassume A3:
G = a * F
; :: thesis: for x being Element of X holds g . x = a * (f . x)

let x be Element of X; :: thesis: g . x = a * (f . x)

g1 = a * f1 by A2, A3, Th3;

hence g . x = a * (f . x) by A1, CFUNCDOM:4; :: thesis: verum

end;let x be Element of X; :: thesis: g . x = a * (f . x)

g1 = a * f1 by A2, A3, Th3;

hence g . x = a * (f . x) by A1, CFUNCDOM:4; :: thesis: verum

then g1 = a * f1 by A1, CFUNCDOM:4;

hence G = a * F by A2, Th3; :: thesis: verum