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;
A1: [:dom f,dom g:] = dom [:f,g:] by Def8;
    thus q in [:f,g:].:[:B,A:] implies q in [:f.:B,g.:A:]
    proof
      assume q in [:f,g:].:[:B,A:];
      then consider p being object such that
A2:   p in dom [:f,g:] and
A3:   p in [:B,A:] and
A4:   q = [:f,g:].p by FUNCT_1:def 6;
      dom [:f,g:] = [:dom f,dom g:] by Def8;
      then consider x,y being object such that
A5:   x in dom f and
A6:   y in dom g and
A7:   p =[x,y] by A2,ZFMISC_1:def 2;
      x in B by A3,A7,ZFMISC_1:87;
      then
A8:   f.x in f.:B by A5,FUNCT_1:def 6;
      y in A by A3,A7,ZFMISC_1:87;
      then
A9:   g.y in g.:A by A6,FUNCT_1:def 6;
      q = [:f,g:].(x,y) by A4,A7;
      then q=[f.x,g.y] by A5,A6,Def8;
      hence thesis by A8,A9,ZFMISC_1:87;
    end;
    assume q in [:f.:B,g.:A:];
    then consider y1,y2 being object such that
A10: y1 in f.:B and
A11: y2 in g.:A and
A12: q = [y1,y2] by ZFMISC_1:def 2;
    consider x1 being object such that
A13: x1 in dom f and
A14: x1 in B and
A15: y1 = f.x1 by A10,FUNCT_1:def 6;
    consider x2 being object such that
A16: x2 in dom g and
A17: x2 in A and
A18: y2 = g.x2 by A11,FUNCT_1:def 6;
A19: [:f,g:].(x1,x2) = [f.x1,g.x2] by A13,A16,Def8;
    [x1,x2] in [:dom f,dom g:] & [x1,x2] in [:B,A:] by A13,A14,A16,A17,
ZFMISC_1:87;
    hence thesis by A12,A15,A18,A1,A19,FUNCT_1:def 6;
  end;
  hence thesis by TARSKI:2;
end;
