reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem Th21:
  x in Union disjoin f implies ex y,z being object st x = [y,z]
proof
  assume x in Union disjoin f;
  then consider X such that
A1: x in X and
A2: X in rng disjoin f by TARSKI:def 4;
  consider y being object such that
A3: y in dom disjoin f and
A4: X = (disjoin f).y by A2,FUNCT_1:def 3;
  y in dom f by A3,Def3;
  then X = [:f.y,{y}:] by A4,Def3;
  hence thesis by A1,RELAT_1:def 1;
end;
