reserve X,Y,z,s for set, L,L1,L2,A,B for List of X, x for Element of X,
  O,O1,O2,O3 for Operation of X, a,b,y for Element of X, n,m for Nat;

theorem
  dom O1 = X & dom O2 = X implies (O1 OR O2)\& O = (O1 \& O) AND (O2 \& O)
  proof assume
A1: dom O1 = X & dom O2 = X;
    let z,s be object;
    thus [z,s] in (O1 OR O2)\& O implies [z,s] in (O1 \& O) AND (O2 \& O)
    proof
      assume
A2:   [z,s] in (O1 OR O2)\& O; then
      reconsider z,s as Element of X by ZFMISC_1:87;
      s in z.((O1 OR O2)\& O) by A2,RELAT_1:169; then
      s in z.(O1 OR O2)\&O by Th35; then
      s in ((z.O1) OR (z.O2))\&O & z.O1 <> {} & z.O2 <> {}
      by A1,RELAT_1:170,RELSET_2:10; then
      s in (z.O1\&O)AND(z.O2\&O) by Th24; then
      s in (z.(O1\&O))AND(z.O2\&O) by Th35; then
      s in (z.(O1\&O))AND(z.(O2\&O)) by Th35; then
      s in z.((O1\&O)AND(O2\&O)) by RELSET_2:11;
      hence thesis by RELAT_1:169;
    end;
    assume
A3: [z,s] in (O1 \& O) AND (O2 \& O); then
    reconsider z,s as Element of X by ZFMISC_1:87;
    s in z.((O1 \& O) AND (O2 \& O)) by A3,RELAT_1:169; then
    s in (z.(O1\&O))AND(z.(O2\&O)) by RELSET_2:11; then
    s in (z.(O1\&O))AND(z.O2\&O) by Th35; then
    s in (z.O1\&O)AND(z.O2\&O) & z.O1 <> {} & z.O2 <> {}
    by Th35,A1,RELAT_1:170; then
    s in ((z.O1) OR (z.O2))\&O by Th24; then
    s in (z.(O1 OR O2))\&O by RELSET_2:10; then
    s in z.((O1 OR O2)\&O) by Th35;
    hence thesis by RELAT_1:169;
  end;
