 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
for x,y be Point of G.i
 holds reproj(i,0.(product G)).(x-y)
   = reproj(i,0.(product G)).x - reproj(i,0.(product G)).y
proof
   let x,y  be Point of G.i;
   reconsider v=reproj(i,0.(product G)).(x-y) as Element of product carr G
     by Th10;
   reconsider s=reproj(i,0.(product G)).x as Element of product carr G
     by Th10;
   reconsider t=reproj(i,0.(product G)).y as Element of product carr G
     by Th10;

   for j be Element of dom G holds v.j=s.j - t.j
   proof
    let j be Element of dom G;
    per cases;
    suppose A1: i= j; then
     reconsider yy=y as Point of G.j;
     v.j = x-y by Th33,A1; then
     v.j =s.j - yy by Th33,A1;
     hence v.j =s.j - t.j by Th33,A1;
    end;
    suppose A2: i <> j; then
     v.j = 0.(G.j) by Th33; then
     v.j = 0.(G.j) - 0.(G.j) by RLVECT_1:13; then
     v.j =s.j - 0.(G.j) by Th33,A2;
     hence v.j =s.j - t.j by Th33,A2;
    end;
   end;
   hence thesis by Th15;
end;
