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, a being Subset-Family of X holds union union a
  = union { union A where A is Subset of X: A in a }
proof
  let X be set, a be Subset-Family of X;
  thus union union a c= union { union A where A is Subset of X: A in a }
  proof
    let e be object;
    assume e in union union a;
    then consider B being set such that
A1: e in B and
A2: B in union a by TARSKI:def 4;
    consider C being set such that
A3: B in C and
A4: C in a by A2,TARSKI:def 4;
A5: union C in { union A where A is Subset of X: A in a } by A4;
    e in union C by A1,A3,TARSKI:def 4;
    hence thesis by A5,TARSKI:def 4;
  end;
  let e be object;
  assume e in union { union A where A is Subset of X: A in a };
  then consider c being set such that
A6: e in c and
A7: c in { union A where A is Subset of X: A in a } by TARSKI:def 4;
  consider A being Subset of X such that
A8: c = union A and
A9: A in a by A7;
  consider b being set such that
A10: e in b and
A11: b in A by A6,A8,TARSKI:def 4;
  b in union a by A9,A11,TARSKI:def 4;
  hence thesis by A10,TARSKI:def 4;
end;
