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

theorem Th9:
  (FuncAdd(X,Y)). ((FuncExtMult(X,Y)).[a,f],(FuncExtMult(X,Y)).[b,
  f]) = (FuncExtMult(X,Y)).[a+b,f]
proof
 reconsider a,b as Element of REAL by XREAL_0:def 1;
   now
    let x be Element of X;
    thus ((FuncAdd(X,Y)). ((FuncExtMult(X,Y)).[a,f],(FuncExtMult(X,Y)).[b,f]))
    .x = ((FuncExtMult(X,Y)).[a,f]).x + ((FuncExtMult(X,Y)).[b,f]).x by Th1
      .= a*(f.x) + ((FuncExtMult(X,Y)).[b,f]).x by Th2
      .= a*(f.x) + b*(f.x) by Th2
      .= (a+b)*(f.x) by RLVECT_1:def 6
      .= ((FuncExtMult(X,Y)).[a+b,f]).x by Th2;
  end;
  hence thesis by FUNCT_2:63;
end;
