reserve W for Universe,
  H for ZF-formula,
  x,y,z,X for set,
  k for Variable,
  f for Function of VAR,W,
  u,v for Element of W;
reserve F for Function,
  A,B,C for Ordinal,
  a,b,b1,b2,c for Ordinal of W,
  fi for Ordinal-Sequence,
  phi for Ordinal-Sequence of W,
  H for ZF-formula;
reserve psi for Ordinal-Sequence;

theorem Th14:
  for phi being Ordinal-Sequence of W holds union(phi,a) = Union (
  phi|a) & union(phi|a,a) = Union (phi|a)
proof
  let phi be Ordinal-Sequence of W;
  On W c= W by ORDINAL2:7;
  then rng (phi|a) c= rng phi & rng phi c= W by RELAT_1:70;
  then a c= Rank a & phi|a = W|`(phi|a) by CLASSES1:38,RELAT_1:94,XBOOLE_1:1;
  hence thesis by Th13,FUNCT_1:51;
end;
