let X be non empty TopSpace; :: thesis: for a being Real
for x being Point of (R_Normed_Space_of_C_0_Functions X)
for y being Point of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
a * x = a * y
let a be Real; :: thesis: for x being Point of (R_Normed_Space_of_C_0_Functions X)
for y being Point of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
a * x = a * y
let x be Point of (R_Normed_Space_of_C_0_Functions X); :: thesis: for y being Point of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) st x = y holds
a * x = a * y
let y be Point of (R_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
reconsider aa = a as Element of REAL by XREAL_0:def 1;
thus a * x = ( the Mult of (RAlgebra the carrier of X) | [:REAL,(C_0_Functions X):]) . [a,x] by RSSPACE:def 9
.= the Mult of (RAlgebra the carrier of X) . [aa,x] by FUNCT_1:49
.= ( the Mult of (RAlgebra the carrier of X) | [:REAL,(BoundedFunctions the carrier of X):]) . [aa,y] by A1, FUNCT_1:49
.= a * y by C0SP1:def 11 ; :: thesis: verum