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,C:] = f"B /\ g"C
proof
  let f,g be Function;
  for x being object holds x in <:f,g:>"[:B,C:] iff x in f"B /\ g"C
  proof let x be object;
    thus x in <:f,g:>"[:B,C:] implies x in f"B /\ g"C
    proof
      assume
A1:   x in <:f,g:>"[:B,C:];
      then <:f,g:>.x in [:B,C:] by FUNCT_1:def 7;
      then consider y1,y2 being object such that
A2:   y1 in B and
A3:   y2 in C and
A4:   <:f,g:>.x = [y1,y2] by ZFMISC_1:def 2;
A5:   x in dom <:f,g:> by A1,FUNCT_1:def 7;
      then
A6:   x in dom f /\ dom g by Def7;
      then
A7:   x in dom g by XBOOLE_0:def 4;
A8:   [y1,y2] = [f.x,g.x] by A4,A5,Def7;
      then y2 = g.x by XTUPLE_0:1;
      then
A9:   x in g"C by A3,A7,FUNCT_1:def 7;
A10:  x in dom f by A6,XBOOLE_0:def 4;
      y1 =f.x by A8,XTUPLE_0:1;
      then x in f"B by A2,A10,FUNCT_1:def 7;
      hence thesis by A9,XBOOLE_0:def 4;
    end;
    assume
A11: x in f"B /\ g"C;
    then
A12: x in g"C by XBOOLE_0:def 4;
    then
A13: x in dom g by FUNCT_1:def 7;
A14: x in f"B by A11,XBOOLE_0:def 4;
    then x in dom f by FUNCT_1:def 7;
    then
A15: x in dom f /\ dom g by A13,XBOOLE_0:def 4;
    then
A16: x in dom <:f,g:> by Def7;
A17: g.x in C by A12,FUNCT_1:def 7;
    f.x in B by A14,FUNCT_1:def 7;
    then [f.x,g.x] in [:B,C:] by A17,ZFMISC_1:def 2;
    then <:f,g:>.x in [:B,C:] by A15,Th48;
    hence thesis by A16,FUNCT_1:def 7;
  end;
  hence thesis by TARSKI:2;
end;
