 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 Th49:
for G be RealNormSpace-Sequence,
    i,j be Element of dom G, x,y be Point of product G,
    xi be Point of G.i
  st y = reproj(i,x).xi & i <> j holds proj(j).x = proj(j).y
proof
   let G be RealNormSpace-Sequence,
       i,j 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 & i <> j;
   reconsider y1 = y as Element of product carr G by Th10;
A2:y = x +* (i,xi) by A1,Def4;
   set ix = i .--> xi;
A3:  the carrier of (product G) = product carr G by Th10;
   y1.j = x.j by A2,A1,FUNCT_7:32; then
   proj(j).y = x.j by Def3;
   hence thesis by A3,Def3;
end;
