reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;

theorem Th5:
  [:[:X,Y:],Z:],[:X,[:Y,Z:]:] are_equipotent & card [:[:X,Y:],Z:]
  = card [:X,[:Y,Z:]:]
proof
  deffunc f(object) = [$1`1`1,[$1`1`2,$1`2]];
  consider f such that
A1: dom f = [:[:X,Y:],Z:] &
for x being object st x in [:[:X,Y:],Z:] holds f.x = f(x)
  from FUNCT_1:sch 3;
  thus [:[:X,Y:],Z:],[:X,[:Y,Z:]:] are_equipotent
  proof
    take f;
    thus f is one-to-one
    proof
      let x,y be object;
      assume that
A2:   x in dom f and
A3:   y in dom f;
      assume
A4:   f.x = f.y;
A5:   x = [x`1,x`2] & y = [y`1,y`2] by A1,A2,A3,MCART_1:21;
      x`1 in [:X,Y:] by A1,A2,MCART_1:10;
      then
A6:   x`1 = [x`1`1,x`1`2 ] by MCART_1:21;
A7:   f.x = [x`1`1,[x`1`2,x`2]] & f.y = [y`1`1,[y`1`2,y`2]] by A1,A2,A3;
      then
A8:   x`1`1 = y`1`1 by A4,XTUPLE_0:1;
      y`1 in [:X,Y :] by A1,A3,MCART_1:10;
      then
A9:   y`1 = [y`1`1,y`1`2] by MCART_1:21;
A10:  [x`1`2,x`2] = [y`1`2,y`2] by A7,A4,XTUPLE_0:1;
      then x`1`2 = y`1`2 by XTUPLE_0:1;
      hence thesis by A5,A8,A10,A6,A9,XTUPLE_0:1;
    end;
    thus dom f = [:[:X,Y:],Z:] by A1;
    thus rng f c= [:X,[:Y,Z:]:]
    proof
      let x be object;
      assume x in rng f;
      then consider y being object such that
A11:  y in dom f and
A12:  x = f.y by FUNCT_1:def 3;
A13:  y`1 in [:X,Y:] by A1,A11,MCART_1:10;
      then
A14:  y`1`2 in Y by MCART_1:10;
      y`2 in Z by A1,A11,MCART_1:10;
      then
A15:  [y`1`2,y`2] in [:Y,Z:] by A14,ZFMISC_1:87;
A16:  y`1`1 in X by A13,MCART_1:10;
      x = [y`1`1,[y`1`2,y`2]] by A1,A11,A12;
      hence thesis by A16,A15,ZFMISC_1:87;
    end;
    let x be object;
A17: [x`1,x`2`1]`1 = x`1 & [x`1,x`2`1]`2 = x`2 `1;
A18: [[x`1,x`2`1],x`2`2]`1 = [x`1,x`2`1] & [[x`1,x`2`1],x`2`2]`2 = x`2`2;
    assume
A19: x in [:X,[:Y,Z:]:];
    then
A20: x`2 in [:Y,Z:] by MCART_1:10;
    then
A21: x`2`1 in Y by MCART_1:10;
A22: x`2`2 in Z by A20,MCART_1:10;
    x`1 in X by A19,MCART_1:10;
    then
A23: [x`1,x`2`1] in [:X,Y:] by A21,ZFMISC_1:87;
    then
A24: [[x`1,x`2`1],x`2`2] in [:[:X,Y:],Z:] by A22,ZFMISC_1:87;
A25: x`2 = [x`2`1,x`2`2] by A20,MCART_1:21;
    x = [x`1,x`2] by A19,MCART_1:21;
    then x = f.[[x`1,x`2`1],x`2`2] by A1,A25,A17,A23,A18,A22,ZFMISC_1:87;
    hence thesis by A1,A24,FUNCT_1:def 3;
  end;
  hence thesis by CARD_1:5;
end;
