reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem
  for f,g being Function holds [:f,g:]"[:B,A:] = [:f"B,g"A:]
proof
  let f,g be Function;
  for q being object holds q in [:f,g:]"[:B,A:] iff q in [:f"B,g"A:]
  proof let q be object;
    thus q in [:f,g:]"[:B,A:] implies q in [:f"B,g"A:]
    proof
      assume
A1:   q in [:f,g:]"[:B,A:];
      then
A2:   [:f,g:].q in [:B,A:] by FUNCT_1:def 7;
      q in dom [:f,g:] by A1,FUNCT_1:def 7;
      then q in [:dom f,dom g:] by Def8;
      then consider x1,x2 being object such that
A3:   x1 in dom f and
A4:   x2 in dom g and
A5:   q = [x1,x2] by ZFMISC_1:def 2;
      [:f,g:].q = [:f,g:].(x1,x2) by A5;
      then
A6:   [f.x1,g.x2] in [:B,A:] by A3,A4,A2,Def8;
      then g.x2 in A by ZFMISC_1:87;
      then
A7:   x2 in g"A by A4,FUNCT_1:def 7;
      f.x1 in B by A6,ZFMISC_1:87;
      then x1 in f"B by A3,FUNCT_1:def 7;
      hence thesis by A5,A7,ZFMISC_1:87;
    end;
    assume q in [:f"B,g"A:];
    then consider x1,x2 being object such that
A8: x1 in f"B & x2 in g"A and
A9: q = [x1,x2] by ZFMISC_1:def 2;
    f.x1 in B & g.x2 in A by A8,FUNCT_1:def 7;
    then
A10: [f.x1,g.x2] in [:B,A:] by ZFMISC_1:87;
    x1 in dom f & x2 in dom g by A8,FUNCT_1:def 7;
    then
A11: [x1,x2] in [:dom f,dom g:] & [:f,g:].(x1,x2) = [f.x1,g.x2] by Def8,
ZFMISC_1:87;
    [:dom f,dom g:] = dom [:f,g:] by Def8;
    hence thesis by A9,A11,A10,FUNCT_1:def 7;
  end;
  hence thesis by TARSKI:2;
end;
