reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th25:
  for X, Y be RealNormSpace for f,h be VECTOR of
  R_VectorSpace_of_BoundedLinearOperators(X,Y)
  for a be Real holds h = a*f iff
  for x be VECTOR of X holds h.x = a * f.x
proof
  let X, Y be RealNormSpace;
  let f,h be VECTOR of R_VectorSpace_of_BoundedLinearOperators(X,Y);
  let a be Real;
A1: R_VectorSpace_of_BoundedLinearOperators(X,Y) is Subspace of
  R_VectorSpace_of_LinearOperators(X,Y) by Th22,RSSPACE:11;
  then reconsider f1=f as VECTOR of R_VectorSpace_of_LinearOperators(X,Y) by
RLSUB_1:10;
  reconsider h1=h as VECTOR of R_VectorSpace_of_LinearOperators(X,Y) by A1,
RLSUB_1:10;
  hereby
    assume
A2: h = a*f;
    let x be Element of X;
    h1=a*f1 by A1,A2,RLSUB_1:14;
    hence h.x=a*f.x by Th17;
  end;
  assume for x be Element of X holds h.x=a*f.x;
  then h1=a*f1 by Th17;
  hence thesis by A1,RLSUB_1:14;
end;
