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;
