theorem Th24:
  a1 is_a_left_unity_wrt f1 & a2 is_a_left_unity_wrt f2 iff [a1,a2
  ] is_a_left_unity_wrt |:f1,f2:|
proof
  thus a1 is_a_left_unity_wrt f1 & a2 is_a_left_unity_wrt f2 implies [a1,a2]
  is_a_left_unity_wrt |:f1,f2:|
  proof
    defpred P[set] means |:f1,f2:|.([a1,a2],$1) = $1;
    assume
A1: f1.(a1,b1) = b1;
    assume
A2: f2.(a2,b2) = b2;
A3: now
      let b1,b2;
      |:f1,f2:|.([a1,a2],[b1,b2]) = [f1.(a1,b1),f2.(a2,b2)] by Th21
        .= [b1,f2.(a2,b2)] by A1
        .= [b1,b2] by A2;
      hence P[[b1,b2]];
    end;
    thus for a being Element of [:D1,D2:] holds P[a] from AuxCart1(A3);
  end;
  assume
A4: for a being Element of [:D1,D2:] holds |:f1,f2:|.([a1,a2],a) = a;
  thus f1.(a1,b1) = b1
  proof
    set b2 = the Element of D2;
    [f1.(a1,b1),f2.(a2,b2)] = |:f1,f2:|.([a1,a2],[b1,b2]) by Th21
      .= [b1,b2] by A4;
    hence thesis by XTUPLE_0:1;
  end;
  set b1 = the Element of D1;
  let b2;
  [f1.(a1,b1),f2.(a2,b2)] = |:f1,f2:|.([a1,a2],[b1,b2]) by Th21
    .= [b1,b2] by A4;
  hence thesis by XTUPLE_0:1;
end;
