let X, Y be RealNormSpace; ( R_NormSpace_of_BoundedLinearOperators (X,Y) is reflexive & R_NormSpace_of_BoundedLinearOperators (X,Y) is discerning & R_NormSpace_of_BoundedLinearOperators (X,Y) is RealNormSpace-like )
thus
||.(0. (R_NormSpace_of_BoundedLinearOperators (X,Y))).|| = 0
by Th37; NORMSP_0:def 6 ( R_NormSpace_of_BoundedLinearOperators (X,Y) is discerning & R_NormSpace_of_BoundedLinearOperators (X,Y) is RealNormSpace-like )
for x, y being Point of (R_NormSpace_of_BoundedLinearOperators (X,Y))
for a being Real holds
( ( ||.x.|| = 0 implies x = 0. (R_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( x = 0. (R_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
by Th37;
hence
( R_NormSpace_of_BoundedLinearOperators (X,Y) is discerning & R_NormSpace_of_BoundedLinearOperators (X,Y) is RealNormSpace-like )
by NORMSP_1:def 1; verum