
theorem LMLT11:
  (Int-mult-left(F_Rat)) | [:INT, INT:] = Int-mult-left(INT.Ring)
  proof
    set ad = (Int-mult-left(F_Rat)) | [:INT, INT:];
    [:INT,INT:] c= [:INT,RAT:] by NUMBERS:14,ZFMISC_1:96;
    then A2: [:INT,INT:] c= dom (Int-mult-left(F_Rat)) by FUNCT_2:def 1;
    A3: dom (Int-mult-left(INT.Ring)) = [:INT,INT:] by FUNCT_2:def 1;
    for z being object st z in dom ad holds ad.z = (Int-mult-left(INT.Ring)).z
    proof
      let z be object;
      assume A4: z in dom ad;
      then consider x, y be object such that
      A5: x in INT & y in INT & z = [x,y] by ZFMISC_1:def 2;
      reconsider x1 = x, y1 = y as Element of INT.Ring by A5;
      reconsider y2 = y1 as Element of F_Rat by RAT_1:def 2;
      reconsider y3 = y1 as Element of INT.Ring;
      thus ad.z = (Int-mult-left(F_Rat)). (x1,y1) by A4,A5,FUNCT_1:49
      .= x1*y2 by ZMODUL07:15
      .= (Int-mult-left(INT.Ring)). (x1,y3) by ZMODUL06:14
      .= (Int-mult-left(INT.Ring)).z by A5;
    end;
    hence thesis by A2,A3,FUNCT_1:2,RELAT_1:62;
  end;
