
theorem Th5:
  multrat||INT = multint
  proof
    set mu = multrat||INT;
    [:INT,INT:] c= [:RAT,RAT:] by NUMBERS:14,ZFMISC_1:96;
    then
A1: [:INT,INT:] c= dom(multrat) by FUNCT_2:def 1;
    then
A2: dom mu = [:INT,INT:] by RELAT_1:62;
A3: dom(multint) = [:INT,INT:] by FUNCT_2:def 1;
    for z be object st z in dom mu holds mu.z = multint.z
    proof
      let z be object;
    assume
A4: z in dom mu;
    then consider x, y be object such that
A5: x in INT & y in INT & z = [x,y] by A2,ZFMISC_1:def 2;
    reconsider x1 = x, y1 = y as Integer by A5;
    thus mu.z = multrat.(x1,y1) by A4,A5,A2,FUNCT_1:49
    .= x1*y1 by BINOP_2:def 17
    .= multint.(x1,y1) by BINOP_2:def 22
    .= multint.z by A5;
  end;
  hence thesis by A1,A3,RELAT_1:62;
end;
