 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 Th38:
for x be Point of G.i st x <> 0.(G.i)
 holds reproj(i,0.(product G)).x <> 0.(product G)
proof
   let x be Point of G.i;
   assume A1: x <> 0.(G.i);
   assume A2: reproj(i,0.(product G)).x = 0.(product G);
   reconsider v=reproj(i,0.(product G)).x as Element of product carr G
     by Th10;
   x = v.i by Th33;
   hence contradiction by A1,Th14,A2;
end;
