reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem
  for A being set, F being Subset-Family of A, R be Relation holds
  R.: union F = union {R.:X where X is Subset of A: X in F} :: (3.1.1)
proof
  let A be set, F be Subset-Family of A, R be Relation;
  thus R.: union F c= union {R.:f where f is Subset of A: f in F}
  proof
    let y be object;
    assume y in R.:(union F);
    then consider x being object such that
A1: [x,y] in R and
A2: x in union F by RELAT_1:def 13;
    consider Y being set such that
A3: x in Y and
A4: Y in F by A2,TARSKI:def 4;
    set Z = R.:Y;
A5: y in Z by A1,A3,RELAT_1:def 13;
    Z in {R.:f where f is Subset of A: f in F} by A4;
    hence thesis by A5,TARSKI:def 4;
  end;
  let y be object;
  assume y in union {R.:f where f is Subset of A: f in F};
  then consider Y being set such that
A6: y in Y and
A7: Y in {R.:f where f is Subset of A: f in F} by TARSKI:def 4;
  consider f being Subset of A such that
A8: Y = R.:f and
A9: f in F by A7;
  consider x being object such that
A10: [x,y] in R and
A11: x in f by A6,A8,RELAT_1:def 13;
  x in union F by A9,A11,TARSKI:def 4;
  hence thesis by A10,RELAT_1:def 13;
end;
