
theorem Th1:
  for x being set, D being non empty set holds x /\ union D = union
  the set of all x /\ d where d is Element of D
proof
  let x be set, D be non empty set;
  hereby
    let a be object;
    assume
A1: a in x /\ union D;
    then a in union D by XBOOLE_0:def 4;
    then consider Z being set such that
A2: a in Z and
A3: Z in D by TARSKI:def 4;
A4: x /\ Z in the set of all x /\ d where d is Element of D by A3;
    a in x by A1,XBOOLE_0:def 4;
    then a in x /\ Z by A2,XBOOLE_0:def 4;
    hence
    a in union the set of all x /\ d where d is Element of D by A4,
TARSKI:def 4;
  end;
  let a be object;
  assume a in union the set of all x /\ d where d is Element of D;
  then consider Z being set such that
A5: a in Z and
A6: Z in the set of all x /\ d where d is Element of D by TARSKI:def 4;
  consider d being Element of D such that
A7: Z = x /\ d and
  not contradiction by A6;
  a in d by A5,A7,XBOOLE_0:def 4;
  then
A8: a in union D by TARSKI:def 4;
  a in x by A5,A7,XBOOLE_0:def 4;
  hence thesis by A8,XBOOLE_0:def 4;
end;
