reserve X for non empty set;
reserve Y for ComplexLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Complex;
reserve u,v,w for VECTOR of CLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th8:
  (FuncExtMult(X,Y)).[a,(FuncExtMult(X,Y)).[b,f]] = (FuncExtMult(X, Y)).[a*b,f]
proof
  reconsider a1=a,b1=b, ab = a*b as Element of COMPLEX by XCMPLX_0:def 2;
  now
    let x be Element of X;
    thus ((FuncExtMult(X,Y)).[a1,(FuncExtMult(X,Y)).[b1,f]]).x =
    a1*(((FuncExtMult(X,Y)).[b1,f]).x) by Th2
      .= a*(b*(f.x)) by Th2
      .= (a*b)*(f.x) by CLVECT_1:def 4
      .= ((FuncExtMult(X,Y)).[ab,f]).x by Th2;
  end;
  then
  (FuncExtMult(X,Y)).[a,(FuncExtMult(X,Y)).[b,f]] = (FuncExtMult(X, Y)).[ab,f]
     by FUNCT_2:63;
  hence thesis;
end;
