 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;

theorem Th8:
for S be RealNormSpace, x be Point of S, N1,N2 be Neighbourhood of x holds
    N1/\ N2 is Neighbourhood of x
proof
   let S be RealNormSpace, x be Point of S,
       N1,N2 be Neighbourhood of x;

   consider N be Neighbourhood of x such that
A1: N c= N1 & N c= N2 by NDIFF_1:1;
A2:N c= N1/\ N2 by A1,XBOOLE_1:19;
   consider g be Real such that
A3: 0 < g and
A4: {y where y is Point of S: ||.y-x .|| < g} c= N by NFCONT_1:def 1;
   {y where y is Point of S: ||.y-x .|| < g} c= N1 /\ N2 by A2,A4;
   hence thesis by A3,NFCONT_1:def 1;
end;
