 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);
reserve u,v,w for VECTOR of CLSp_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,A,B for Element of S,
  f,g,h,f1,g1 for PartFunc of X,COMPLEX;
reserve v,u for VECTOR of CLSp_L1Funct M;

theorem Th20:
  f=u implies a(#)f=a*u
proof
  reconsider u2=u as VECTOR of CLSp_PFunctX by TARSKI:def 3;
  reconsider h = a*u2 as Element of PFuncs(X,COMPLEX);
  assume
A1: f=u;
A2: a*u2 = (multcomplexcpfunc X).(a,u2);
A3: dom h = dom f by A1,A2,Th7;
A4:for x be object st x in dom h holds h.x = a*(f.x) by A1,A2,A3,Th7;
A5: h= a(#)f by A3,A4,VALUED_1:def 5;
  reconsider a as Element of COMPLEX by XCMPLX_0:def 2;
  [a,u] in [:COMPLEX,(L1_CFunctions M):];
  hence thesis by A5,FUNCT_1:49;
end;
