reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;
reserve v,u for VECTOR of RLSp_L1Funct M;

theorem Th26:
  f=u implies a(#)f=a*u
proof
  reconsider u2=u as VECTOR of RLSp_PFunctX 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 Th9;
  then for x be object st x in dom h holds h.x = a*(f.x) by A1,Th9;
  then h= a(#)f by A2,VALUED_1:def 5;
  hence thesis by Th5;
end;
