let X be non empty TopSpace; :: thesis: for a being Complex
for x being Point of (C_Normed_Space_of_C_0_Functions X)
for y being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
a * x = a * y
let a be Complex; :: thesis: for x being Point of (C_Normed_Space_of_C_0_Functions X)
for y being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
a * x = a * y
let x be Point of (C_Normed_Space_of_C_0_Functions X); :: thesis: for y being Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
a * x = a * y
let y be Point of (C_Normed_Algebra_of_BoundedFunctions the carrier of X); :: thesis: ( x = y implies a * x = a * y )
assume A1: x = y ; :: thesis: a * x = a * y
thus a * x = ( the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(CC_0_Functions X):]) . [a,x] by CSSPACE:def 9
.= the Mult of (CAlgebra the carrier of X) . [a,x] by FUNCT_1:49
.= ( the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(ComplexBoundedFunctions the carrier of X):]) . [a,y] by FUNCT_1:49, A1
.= a * y by CC0SP1:def 3 ; :: thesis: verum