 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;

theorem Th11:
for i be Element of dom G, r be set, x be Function
  st r in the carrier of (G.i) & x in product carr G
holds x +* (i,r) in the carrier of product G
proof
   let i be Element of dom G, r be set, x be Function;
   assume A1: r in the carrier of (G.i) & x in product carr G; then
   consider g be Function such that
A2:   x = 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;
   set h = x +* (i,r);
   set s = i .--> r;
   s = {i} --> r by FUNCOP_1:def 9; then
A3:dom s = {i};
A4:dom h = dom carr G by A2,FUNCT_7:30;
   for j be object st j in dom carr G holds h.j in (carr G).j
   proof
    let j be object;
    assume A5: j in dom carr G;
    per cases;
    suppose not j in dom s; then
     j <> i by A3,TARSKI:def 1; then
     h.j = x.j by FUNCT_7:32;
     hence h.j in (carr G).j by A2,A5;
    end;
    suppose j in dom s; then
A6:  j = i by TARSKI:def 1; then
     h.j = r by A5,A2,FUNCT_7:31;
     hence h.j in (carr G).j by A1,A6,PRVECT_1:def 11;
    end;
   end; then
   x +* (i,r) in product carr G by A4,CARD_3:def 5;
   hence thesis by Th10;
end;
