 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;
reserve G for RealNormSpace-Sequence;
reserve F for RealNormSpace;
reserve i for Element of dom G;
reserve f,f1,f2 for PartFunc of product G, F;
reserve x for Point of product G;
reserve X for set;

theorem Th33:
 for i,j be Element of dom G,
     x be Point of G.i,
     z be Element of product carr G
      st z= reproj(i,0.(product G)).x
holds
  (i = j implies z.j = x) & (i <> j implies z.j = 0.(G.j))
proof
   let i,j be Element of dom G,
       x be Point of G.i,
       z be Element of product carr G;
   assume A1: z= reproj(i,0.(product G)).x;
   reconsider Zr = 0.(product G) as Element of product carr G by Th10;
   reconsider ixr = (reproj(i,0.(product G))).x
             as Element of product carr G by Th10;
A2:reproj(i,0.(product G)).x = 0.(product G) +* (i,x) by Def4;
   set ix= i .--> x;
   consider g be Function such that
A3:   Zr = g & dom g = dom carr G
    & for i be object st i in dom carr G holds g.i in (carr G).i
        by CARD_3:def 5;
A4:dom Zr = dom G by A3,Lm1;
   now assume i <> j; then
    z.j = Zr.j by A1,A2,FUNCT_7:32;
    hence z.j = 0.(G.j) by Th14;
   end;
   hence thesis by A1,A2,A4,FUNCT_7:31;
end;
