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 Th7:
  (FuncExtMult(X,Y)).[1,f] = f
proof
  now
    let x be Element of X;
    thus ((FuncExtMult(X,Y)).[jj,f]).x = 1*(f.x) by Th2
      .= f.x by RLVECT_1:def 8;
  end;
  hence thesis by FUNCT_2:63;
end;
