reserve X for set, R,R1,R2 for Relation;
reserve x,y,z for set;
reserve n,m,k for Nat;

theorem Th10:
  field (R1\,R2) = field R1 \/ field R2
  proof
    thus field (R1\,R2) c= field R1 \/ field R2
    proof
      let z be object; assume z in field (R1\,R2); then
      z in dom (R1\,R2) or z in rng (R1\,R2) by XBOOLE_0:def 3; then
      consider y be object such that
A1:   [z,y] in R1\,R2 or [y,z] in R1\,R2 by XTUPLE_0:def 12,def 13;
      reconsider zz = z, y as set by TARSKI:1;
      zz,y in R1\,R2 or y,zz in R1\,R2 by A1; then
      zz,y in R1 or y,zz in R1 or zz,y in R2 or y,zz in R2 by Th9; then
      [z,y] in R1 or [y,z] in R1 or [z,y] in R2 or [y,z] in R2; then
      z in field R1 or z in field R2 by RELAT_1:15;
      hence thesis by XBOOLE_0:def 3;
    end;
    let z be object;
    assume z in field R1 \/ field R2; then
    z in field R1 or z in field R2 by XBOOLE_0:def 3; then
    z in dom R1 or z in rng R1 or z in dom R2 or z in rng R2 by XBOOLE_0:def 3;
    then
    consider y be object such that
A2: [z,y] in R1 or [y,z] in R1 or [z,y] in R2 or [y,z] in R2
    by XTUPLE_0:def 12,def 13;
    reconsider zz = z, y as set by TARSKI:1;
    zz,y in R1 or y,zz in R1 or zz,y in R2 & y,zz nin R1 or y,zz in R1 or
    y,zz in R2 & zz,y nin R1 or zz,y in R1 by A2; then
    zz,y in R1\,R2 or y,zz in R1\,R2 by Th9; then
    [z,y] in R1\,R2 or [y,z] in R1\,R2;
    hence thesis by RELAT_1:15;
  end;
