 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 Th46:
for G be RealNormSpace-Sequence,
    i be Element of dom G,
    x,y be Point of product G,
    xi be Point of G.i
  st y = reproj(i,x).xi holds proj(i).y = xi
proof
   let G be RealNormSpace-Sequence,
       i be Element of dom G, x,y be Point of product G,
       xi be Point of G.i;
   assume A1: y = reproj(i,x).xi;
A2:y = x +* (i,xi) by A1,Def4;
   x in the carrier of product G; then
   x in product carr G by Th10; then
   consider g being Function such that
A3:x = g & dom g = dom carr G &
    for y be object st y in dom carr G holds g.y in (carr G).y
by CARD_3:def 5;
A4:i in dom G;
A5:i in dom x by Lm1,A4,A3;
   y is Element of product carr G by Th10; then
   proj(i).y = (x +* (i,xi)).i by A2,Def3;
   hence proj(i).y = xi by A5,FUNCT_7:31;
end;
