reserve X for set, a,b,c,x,y,z for object;
reserve P,R for Relation;

theorem
  R is strongly_connected iff [:field R, field R:] = R \/ R~
proof
  hereby
    assume
A1: R is strongly_connected;
    now
      let x be object;
A2:   now
        assume
A3:     x in R \/ R~;
        then consider y,z being object such that
A4:     x = [y,z] by RELAT_1:def 1;
        [y,z] in R or [y,z] in R~ by A3,A4,XBOOLE_0:def 3;
        then [y,z] in R or [z,y] in R by RELAT_1:def 7;
        then y in field R & z in field R by RELAT_1:15;
        hence x in [:field R, field R:] by A4,ZFMISC_1:87;
      end;
      now
        assume x in [:field R, field R:];
        then consider y,z being object such that
A5:     y in field R & z in field R and
A6:     x = [y,z] by ZFMISC_1:def 2;
        [y,z] in R or [z,y] in R by A1,A5,Def7;
        then [y,z] in R or [y,z] in R~ by RELAT_1:def 7;
        hence x in R \/ R~ by A6,XBOOLE_0:def 3;
      end;
      hence x in [:field R, field R:] iff x in R \/ R~ by A2;
    end;
    hence [:field R, field R:] = R \/ R~;
  end;
  assume
A7: [:field R, field R:] = R \/ R~;
  let a,b;
  a in field R & b in field R implies [a,b] in R \/ R~ by A7,ZFMISC_1:87;
  then a in field R & b in field R implies [a,b] in R or [a,b] in R~
  by XBOOLE_0:def 3;
  hence thesis by RELAT_1:def 7;
end;
