
theorem Th2:
  for A,B being non empty set holds [:union A, union B:] = union {
  [:a,b:] where a is Element of A, b is Element of B : a in A & b in B }
proof
  let A,B be non empty set;
  set Y = { [:a,b:] where a is Element of A, b is Element of B : a in A & b in
  B };
  thus [:union A, union B:] c= union Y
  proof
    let z be object;
    assume
A1: z in [:union A, union B:];
    then consider x,y being object such that
A2: z = [x,y] by RELAT_1:def 1;
    y in union B by A1,A2,ZFMISC_1:87;
    then consider b9 being set such that
A3: y in b9 and
A4: b9 in B by TARSKI:def 4;
    x in union A by A1,A2,ZFMISC_1:87;
    then consider a9 being set such that
A5: x in a9 and
A6: a9 in A by TARSKI:def 4;
    reconsider b9 as Element of B by A4;
    reconsider a9 as Element of A by A6;
A7: [:a9,b9:] in Y;
    z in [:a9,b9:] by A2,A5,A3,ZFMISC_1:def 2;
    hence thesis by A7,TARSKI:def 4;
  end;
  let z be object;
  assume z in union Y;
  then consider e being set such that
A8: z in e and
A9: e in Y by TARSKI:def 4;
  consider a9 being Element of A, b9 being Element of B such that
A10: [:a9,b9:] = e and
  a9 in A and
  b9 in B by A9;
  consider x,y being object such that
A11: x in a9 & y in b9 and
A12: z = [x,y] by A8,A10,ZFMISC_1:def 2;
  x in union A & y in union B by A11,TARSKI:def 4;
  hence thesis by A12,ZFMISC_1:def 2;
end;
