reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;
reserve X,Y,Z for non empty set;

theorem
  for X being set, D being a_partition of X, A,B being Subset of D
  holds union(A /\ B) = union A /\ union B
proof
  let X be set, D be a_partition of X, A,B be Subset of D;
  thus union(A/\B) c= union A /\ union B by ZFMISC_1:79;
  let e be object;
  assume
A1: e in union A /\ union B;
  then e in union A by XBOOLE_0:def 4;
  then consider a being set such that
A2: e in a and
A3: a in A by TARSKI:def 4;
A4: a in D by A3;
  e in union B by A1,XBOOLE_0:def 4;
  then consider b being set such that
A5: e in b and
A6: b in B by TARSKI:def 4;
A7: b in D by A6;
  not a misses b by A2,A5,XBOOLE_0:3;
  then a = b by A4,A7,Def4;
  then a in A/\B by A3,A6,XBOOLE_0:def 4;
  hence thesis by A2,TARSKI:def 4;
end;
