
theorem Th4:
  addrat||INT = addint
  proof
    set ad = addrat||INT;
    [:INT,INT:] c= [:RAT,RAT:] by NUMBERS:14,ZFMISC_1:96;
    then
A1: [:INT,INT:] c= dom(addrat) by FUNCT_2:def 1;
    then
A2: dom ad = [:INT,INT:] by RELAT_1:62;
A3: dom(addint) = [:INT,INT:] by FUNCT_2:def 1;
    for z be object st z in dom ad holds ad.z = addint.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 A2,ZFMISC_1:def 2;
      reconsider x1 = x, y1 = y as Integer by A5;
      thus ad.z = addrat.(x1,y1) by A4,A5,A2,FUNCT_1:49
      .= x1+y1 by BINOP_2:def 15
      .= addint.(x1,y1) by BINOP_2:def 20
      .= addint.z by A5;
    end;
    hence thesis by A1,A3,RELAT_1:62;
  end;
