 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 Th39:
for x be Point of G.i, a be Real
 holds reproj(i,0.(product G)).(a*x)
   = a*(reproj(i,0.(product G)).x)
proof
   let x  be Point of G.i,a be Real;
    reconsider a as Real;
   reconsider v=reproj(i,0.(product G)).(a*x) as Element of product carr G
     by Th10;
   reconsider s=reproj(i,0.(product G)).x as Element of product carr G
     by Th10;

   for j be Element of dom G holds v.j=a*(s.j)
   proof
    let j be Element of dom G;
    per cases;
    suppose A1: i= j; then
     v.j = a*x by Th33;
     hence v.j =a*(s.j) by Th33,A1;
    end;
    suppose A2: i <> j; then
     v.j = 0.(G.j) by Th33; then
     v.j = a*0.(G.j) by RLVECT_1:10;
     hence v.j =a*(s.j) by Th33,A2;
    end;
   end;
   hence thesis by Th13;
end;
