reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;

theorem Th1:
  for x0 be Point of S for N1,N2 being Neighbourhood of x0 ex N
  being Neighbourhood of x0 st N c= N1 & N c= N2
proof
  let x0 be Point of S;
  let N1,N2 be Neighbourhood of x0;
  consider g1 be Real such that
A1: 0<g1 and
A2: {y where y is Point of S: ||.y-x0 .|| < g1} c= N1 by NFCONT_1:def 1;
  consider g2 be Real such that
A3: 0<g2 and
A4: {y where y is Point of S: ||.y-x0 .|| < g2} c= N2 by NFCONT_1:def 1;
  set g = min(g1,g2);
  take {y where y is Point of S: ||.y-x0 .|| < g};
A5: g<=g2 by XXREAL_0:17;
A6: {y where y is Point of S: ||.y-x0 .|| < g} c= {y where y is Point of S:
  ||.y-x0 .|| < g2}
  proof
    let z be object;
    assume z in {y where y is Point of S: ||.y-x0 .|| < g};
    then consider y be Point of S such that
A7: z=y and
A8: ||.y-x0 .|| < g;
    ||.y-x0 .|| < g2 by A5,A8,XXREAL_0:2;
    hence thesis by A7;
  end;
A9: g<=g1 by XXREAL_0:17;
A10: {y where y is Point of S: ||.y-x0 .|| < g} c= {y where y is Point of S:
  ||.y-x0 .|| < g1}
  proof
    let z be object;
    assume z in {y where y is Point of S: ||.y-x0 .|| < g};
    then consider y be Point of S such that
A11: z=y and
A12: ||.y-x0 .|| < g;
    ||.y-x0 .|| < g1 by A9,A12,XXREAL_0:2;
    hence thesis by A11;
  end;
  0<g by A1,A3,XXREAL_0:15;
  hence thesis by A2,A4,A10,A6,NFCONT_1:3;
end;
