let X be RealNormSpace-Sequence; :: thesis: for Y being RealNormSpace
for f, g, h being VECTOR of (R_VectorSpace_of_MultilinearOperators (X,Y)) holds
( h = f + g iff for x being VECTOR of (product X) holds h . x = (f . x) + (g . x) )
let Y be RealNormSpace; :: thesis: for f, g, h being VECTOR of (R_VectorSpace_of_MultilinearOperators (X,Y)) holds
( h = f + g iff for x being VECTOR of (product X) holds h . x = (f . x) + (g . x) )
let f, g, h be VECTOR of (R_VectorSpace_of_MultilinearOperators (X,Y)); :: thesis: ( h = f + g iff for x being VECTOR of (product X) holds h . x = (f . x) + (g . x) )
reconsider f9 = f, g9 = g, h9 = h as MultilinearOperator of X,Y by Def6;
A1: R_VectorSpace_of_MultilinearOperators (X,Y) is Subspace of RealVectSpace ( the carrier of (product X),Y) by RSSPACE:11;
then reconsider f1 = f, h1 = h, g1 = g as VECTOR of (RealVectSpace ( the carrier of (product X),Y)) by RLSUB_1:10;
A2: now :: thesis: ( h = f + g implies for x being Element of (product X) holds h9 . x = (f9 . x) + (g9 . x) )
assume A3: h = f + g ; :: thesis: for x being Element of (product X) holds h9 . x = (f9 . x) + (g9 . x)
let x be Element of (product X); :: thesis: h9 . x = (f9 . x) + (g9 . x)
h1 = f1 + g1 by A1, A3, RLSUB_1:13;
hence h9 . x = (f9 . x) + (g9 . x) by LOPBAN_1:1; :: thesis: verum
end;
now :: thesis: ( ( for x being Element of (product X) holds h9 . x = (f9 . x) + (g9 . x) ) implies h = f + g )
assume for x being Element of (product X) holds h9 . x = (f9 . x) + (g9 . x) ; :: thesis: h = f + g
then h1 = f1 + g1 by LOPBAN_1:1;
hence h = f + g by A1, RLSUB_1:13; :: thesis: verum
end;
hence ( h = f + g iff for x being VECTOR of (product X) holds h . x = (f . x) + (g . x) ) by A2; :: thesis: verum