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 Th33:
  card(X \/ Y) c= card X +` card Y
proof
  consider f such that
A1: dom f = plus(X,Y) &
for x being object st x in plus(X,Y) holds f.x = g(x) from
  FUNCT_1:sch 3;
  X \/ Y c= rng f
  proof
    let x be object;
    assume x in X \/ Y;
    then
A2: x in X or x in Y by XBOOLE_0:def 3;
    per cases;
    suppose
      x in X;
      then [x,0] in [:X,{0}:] by ZFMISC_1:106;
      then
A3:   [x,0] in plus(X,Y) by XBOOLE_0:def 3;
      [x,0]`1 = x;
      then x = f.[x,0] by A1,A3;
      hence thesis by A1,A3,FUNCT_1:def 3;
    end;
    suppose
      not x in X;
      then [x,1] in [:Y,{1}:] by A2,ZFMISC_1:106;
      then
A4:   [x,1] in plus(X,Y) by XBOOLE_0:def 3;
      [x,1]`1 = x;
      then x = f.[x,1] by A1,A4;
      hence thesis by A1,A4,FUNCT_1:def 3;
    end;
  end;
  then card(X \/ Y) c= card plus(X,Y) by A1,CARD_1:12;
  hence thesis by Th16;
end;
