reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);

theorem Th30:
f = u implies a(#)f = a*u
proof
   reconsider u2=u as VECTOR of RLSp_PFunct X by TARSKI:def 3;
   reconsider h = a*u2 as Element of PFuncs(X,REAL);
   assume A1:f=u; then
A2:dom h = dom f by LPSPACE1:9; then
   for x be object st x in dom h holds h.x = a*(f.x) by A1,LPSPACE1:9; then
   h= a(#)f by A2,VALUED_1:def 5;
   hence thesis by LPSPACE1:5;
end;
