
theorem ConField:
  for R being Relation holds
    R is connected iff [:field R,field R:] = R \/ R~ \/ id (field R)
  proof
    let R be Relation;
    hereby assume
      R is connected; then
      [:field R,field R:] \ id (field R) c= R \/ R~ by RELAT_2:28; then
Z0:   [:field R,field R:] c= R \/ R~ \/ id (field R) by XBOOLE_1:44;
C1:   R c= [:dom R, rng R:] by RELAT_1:7;
      dom (R~) = rng R & rng (R~) = dom R by RELAT_1:20; then
C2:   R~ c= [:rng R, dom R:] by RELAT_1:7;
      set GG = [:rng R,rng R:] \/ [:dom R,dom R:];
      set D = [:dom R,dom R:];
      set RR = [:rng R,rng R:], DR = [:dom R,rng R:];
      set RI = R \/ R~ \/ id field R;
      R \/ R~ c= DR \/ [:rng R, dom R:] by XBOOLE_1:13,C1,C2; then
C4:   RI c= DR \/ [:rng R, dom R:]
        \/ [:field R, field R:] by XBOOLE_1:13;
Z1:   [:field R, field R:] = D \/ [:dom R,rng R:] \/
        [:rng R,dom R:] \/ RR by ZFMISC_1:98; then
      RI c= DR \/ [:rng R, dom R:]
        \/ (DR \/ D \/ ([:rng R,dom R:] \/
          RR)) by C4,XBOOLE_1:4; then
      RI c= [:rng R, dom R:] \/ (DR
        \/ (DR \/ D \/ ([:rng R,dom R:] \/
          RR))) by XBOOLE_1:4; then
      RI c= [:rng R, dom R:] \/ (DR \/ (DR \/ (D \/ ([:rng R,dom R:] \/
          RR)))) by XBOOLE_1:4; then
      RI c= [:rng R, dom R:] \/ (DR
        \/ DR \/ (D \/ ([:rng R,dom R:] \/ RR))) by XBOOLE_1:4; then
      RI c= [:rng R, dom R:] \/ (
        ([:rng R,dom R:] \/ GG) \/ DR) by XBOOLE_1:4; then
      RI c= [:rng R, dom R:] \/ (
        [:rng R,dom R:] \/ (GG \/ DR)) by XBOOLE_1:4; then
      RI c= [:rng R, dom R:] \/
        [:rng R,dom R:] \/ (GG \/ DR) by XBOOLE_1:4; then
      RI c= GG
        \/ (DR \/ [:rng R,dom R:]) by XBOOLE_1:4; then
      RI c= D \/ (DR \/ [:rng R,dom R:])
        \/ RR by XBOOLE_1:4; then
      RI c= [:field R,field R:] by Z1,XBOOLE_1:4;
      hence [:field R,field R:] = R \/ R~ \/ id field R
        by Z0, XBOOLE_0:def 10;
    end;
    assume [:field R,field R:] = R \/ R~ \/ id (field R); then
    [:field R,field R:] \ id (field R) c= R \/ R~ \ id field R
      by XBOOLE_1:40;
    hence thesis by RELAT_2:28,XBOOLE_1:1;
  end;
