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 Th11:
  X1,Y1 are_equipotent & X2,Y2 are_equipotent & x1 <> x2 & y1 <>
y2 implies [:X1,{x1}:] \/ [:X2,{x2}:],[:Y1,{y1}:] \/ [:Y2,{y2}:] are_equipotent
  & card ([:X1,{x1}:] \/ [:X2,{x2}:]) = card ([:Y1,{y1}:] \/ [:Y2,{y2}:])
proof
  assume that
A1: X1,Y1 are_equipotent and
A2: X2,Y2 are_equipotent and
A3: x1 <> x2 and
A4: y1 <> y2;
  {x2},{y2} are_equipotent by CARD_1:28;
  then
A5: [:X2,{x2}:],[:Y2,{y2}:] are_equipotent by A2,Th7;
A6: now
    assume [:Y1,{y1}:] meets [:Y2,{y2}:];
    then consider y being object such that
A7: y in [:Y1,{y1}:] and
A8: y in [:Y2,{y2}:] by XBOOLE_0:3;
    y`2 in {y1} by A7,MCART_1:10;
    then
A9: y`2 = y1 by TARSKI:def 1;
    y`2 in {y2} by A8,MCART_1:10;
    hence contradiction by A4,A9,TARSKI:def 1;
  end;
A10: now
    assume [:X1,{x1}:] meets [:X2,{x2}:];
    then consider x being object such that
A11: x in [:X1,{x1}:] and
A12: x in [:X2,{x2}:] by XBOOLE_0:3;
    x`2 in {x1} by A11,MCART_1:10;
    then
A13: x`2 = x1 by TARSKI:def 1;
    x`2 in {x2} by A12,MCART_1:10;
    hence contradiction by A3,A13,TARSKI:def 1;
  end;
  {x1},{y1} are_equipotent by CARD_1:28;
  then [:X1,{x1}:],[:Y1,{y1}:] are_equipotent by A1,Th7;
  hence [:X1,{x1}:] \/ [:X2,{x2}:],[:Y1,{y1}:] \/ [:Y2,{y2}:]
  are_equipotent by A5,A10,A6,CARD_1:31;
  hence thesis by CARD_1:5;
end;
