reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;

theorem Th25:
  a1 is_a_right_unity_wrt f1 & a2 is_a_right_unity_wrt f2 iff [a1,
  a2] is_a_right_unity_wrt |:f1,f2:|
proof
  thus a1 is_a_right_unity_wrt f1 & a2 is_a_right_unity_wrt f2 implies [a1,a2]
  is_a_right_unity_wrt |:f1,f2:|
  proof
    defpred P[set] means |:f1,f2:|.($1,[a1,a2]) = $1;
    assume
A1: f1.(b1,a1) = b1;
    assume
A2: f2.(b2,a2) = b2;
A3: now
      let b1,b2;
      |:f1,f2:|.([b1,b2],[a1,a2]) = [f1.(b1,a1),f2.(b2,a2)] by Th21
        .= [b1,f2.(b2,a2)] 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:|.(a,[a1,a2]) = a;
  thus f1.(b1,a1) = b1
  proof
    set b2 = the Element of D2;
    [f1.(b1,a1),f2.(b2,a2)] = |:f1,f2:|.([b1,b2],[a1,a2]) by Th21
      .= [b1,b2] by A4;
    hence thesis by XTUPLE_0:1;
  end;
  set b1 = the Element of D1;
  let b2;
  [f1.(b1,a1),f2.(b2,a2)] = |:f1,f2:|.([b1,b2],[a1,a2]) by Th21
    .= [b1,b2] by A4;
  hence thesis by XTUPLE_0:1;
end;
