reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

theorem :: (5)
  R.:^X1 \/ R.:^X2 c= R.:^(X1/\X2)
proof
  let y be object;
  assume
A1: y in R.:^X1 \/ R.:^X2;
A2: {_{X1/\X2}_} = {_{X1}_} /\ {_{X2}_} by Th4;
  then
A3: .:R.:({_{X1/\X2}_}) c= .:R.:{_{X1}_} /\ .:R.:{_{X2}_} by RELAT_1:121;
  per cases by A1,XBOOLE_0:def 3;
  suppose
A4: y in R.:^X1;
    y in Intersect((.:R).:({_{X1/\X2}_}))
    proof
      per cases;
      suppose
A5:     (.:R).:({_{X1/\X2}_}) <> {};
A6:     {_{X1/\X2}_} = {_{X1}_} /\ {_{X2}_} by Th4;
        then
        A7:     (.:R).:({_{X1/\X2}_}) c= (.:R).:{_{X1}_} /\ ((.:R).:{_{X2 }_})
        by RELAT_1:121;
A8:     (.:R).:{_{X1}_} /\ ((.:R).:{_{X2}_}) <> {} by A5,A6,RELAT_1:121
,XBOOLE_1:3;
        (.:R).:{_{X1}_} <> {} by A5,A7;
        then
A9:     y in meet ((.:R).:{_{X1}_}) by A4,SETFAM_1:def 9;
        for Y being set holds
        Y in (.:R).:{_{X1}_} /\ (.:R).:{_{X2}_} implies y in Y
        proof
          let Y be set;
          assume Y in (.:R).:{_{X1}_} /\ (.:R).:{_{X2}_};
          then Y in (.:R).:{_{X1}_} by XBOOLE_0:def 4;
          hence thesis by A9,SETFAM_1:def 1;
        end;
then y in meet ((.:R).:{_{X1}_} /\ (.:R).:{_{X2}_}) by A8,SETFAM_1:def 1;
        then
A10:    y in Intersect ((.:R).:{_{X1}_} /\ (.:R).:{_{X2}_})
        by A8,SETFAM_1:def 9;
        Intersect(.:R.:{_{X1}_} /\ (.:R.:{_{X2}_})) c=
        Intersect(.:R.:({_{X1}_} /\ {_{X2}_})) by RELAT_1:121,SETFAM_1:44;
        hence thesis by A2,A10;
      end;
      suppose (.:R).:({_{X1/\X2}_}) = {};
        then Intersect((.:R).:({_{X1/\X2}_})) = B by SETFAM_1:def 9;
        hence thesis by A4;
      end;
    end;
    hence thesis;
  end;
  suppose
A11: y in R.:^X2;
    y in Intersect((.:R).:({_{X1/\X2}_}))
    proof
      per cases;
      suppose
A12:    (.:R).:({_{X1/\X2}_}) <> {};
        then
A13:    (.:R).:{_{X1}_} /\ ((.:R).:{_{X2}_}) <> {} by A2,RELAT_1:121,XBOOLE_1:3
;
        (.:R).:{_{X2}_} <> {} by A3,A12;
        then
A14:    y in meet ((.:R).:{_{X2}_}) by A11,SETFAM_1:def 9;
        for Y being set holds
        Y in (.:R).:{_{X1}_} /\ (.:R).:{_{X2}_} implies y in Y
        proof
          let Y be set;
          assume Y in (.:R).:{_{X1}_} /\ (.:R).:{_{X2}_};
          then Y in (.:R).:{_{X2}_} by XBOOLE_0:def 4;
          hence thesis by A14,SETFAM_1:def 1;
        end;
then y in meet ((.:R).:{_{X1}_} /\ (.:R).:{_{X2}_}) by A13,SETFAM_1:def 1;
        then
A15:    y in Intersect ((.:R).:{_{X1}_} /\ (.:R).:{_{X2}_})
        by A13,SETFAM_1:def 9;
A16:    {_{X1/\X2}_} = {_{X1}_} /\ {_{X2}_} by Th4;
        Intersect(.:R.:{_{X1}_} /\ (.:R.:{_{X2}_})) c=
        Intersect(.:R.:({_{X1}_} /\ {_{X2}_})) by RELAT_1:121,SETFAM_1:44;
        hence thesis by A15,A16;
      end;
      suppose (.:R).:({_{X1/\X2}_}) = {};
        then Intersect((.:R).:({_{X1/\X2}_})) = B by SETFAM_1:def 9;
        hence thesis by A11;
      end;
    end;
    hence thesis;
  end;
end;
