reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th7:
  for X,Y1,Y2 be set holds UNION(X,Y1\/Y2) = UNION(X,Y1)\/UNION(X,Y2)
proof
  let X,Y1,Y2 be set;
  thus UNION(X,Y1\/Y2) c= UNION(X,Y1)\/UNION(X,Y2)
  proof
    let xy be object such that
A1:   xy in UNION(X,Y1\/Y2);
    consider x,y be set such that
A2:   x in X & y in Y1\/Y2 & xy=x\/y by A1,SETFAM_1:def 4;
    y in Y1 or y in Y2 by A2,XBOOLE_0:def 3;
    then xy in UNION(X,Y1) or xy in UNION(X,Y2) by A2,SETFAM_1:def 4;
    hence thesis by XBOOLE_0:def 3;
  end;
  let xy be object such that
A3: xy in UNION(X,Y1)\/UNION(X,Y2);
  per cases by A3,XBOOLE_0:def 3;
  suppose xy in UNION(X,Y1);
    then consider x,y be set such that
A4:   x in X & y in Y1 & xy=x\/y by SETFAM_1:def 4;
    y in Y1 \/Y2 by A4,XBOOLE_0:def 3;
    hence thesis by A4,SETFAM_1:def 4;
  end;
  suppose xy in UNION(X,Y2);
    then consider x,y be set such that
A5:   x in X & y in Y2 & xy=x\/y by SETFAM_1:def 4;
    y in Y1 \/Y2 by A5,XBOOLE_0:def 3;
    hence thesis by A5,SETFAM_1:def 4;
  end;
end;
