 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 Th40:
for x be Point of product G,a be Real
 holds proj(i).(a*x) = a*(proj(i).x)
proof
   let x be Point of product G,a  be Real;
   reconsider a as Real;
   reconsider v=a*x as Element of product carr G by Th10;
   reconsider s=x as Element of product carr G by Th10;
   proj(i).(a*x)= v.i & proj(i).x = s.i by Def3;
   hence thesis by Th13;
end;
