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;
