 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 Th14:
for G be RealNormSpace-Sequence, p0 be Element of product carr G
holds 0.(product G)=p0
  iff for i be Element of dom G holds p0.i = 0.(G.i)
proof
   let G be RealNormSpace-Sequence,
       p0 be Element of product carr G;
A1:dom carr G = dom G by Lm1;
A2:product G = NORMSTR(# product carr G,zeros G,[:addop G:]
               ,[:multop G:], productnorm G #) by PRVECT_2:6;
   hence 0.(product G) = p0 implies
     for i be Element of dom G holds p0.i = 0.(G.i) by A1,PRVECT_1:def 14;
   assume A3:for i be Element of dom G holds p0.i = 0.(G.i);
   now let i0 be Element of dom carr G;
     reconsider i=i0 as Element of dom G by Lm1;
     p0.i = 0.(G.i) by A3;
     hence p0.i0 = 0.(G.i0);
   end;
   hence 0.(product G)=p0 by A2,PRVECT_1:def 14;
end;
