reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem
  union (A \/ B) = union A \/ union B
proof
A1: union (A \/ B) c= union A \/ union B
  proof
    let x;
    assume x in union (A \/ B);
    then consider Y such that
A2: x in Y and
A3: Y in A \/ B by TARSKI:def 4;
    Y in A or Y in B by A3,XBOOLE_0:def 3;
    then x in union A or x in union B by A2,TARSKI:def 4;
    hence thesis by XBOOLE_0:def 3;
  end;
  union A c= union (A \/ B) & union B c= union (A \/ B) by Th76,XBOOLE_1:7;
  hence thesis by A1,XBOOLE_1:8;
end;
