reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem
  (R1 /\ R2).:{_{X}_} c= R1.:{_{X}_} /\ R2.:{_{X}_}
proof
  let y be object;
  assume y in (R1 /\ R2).:{_{X}_};
  then consider x being object such that
A1: [x,y] in R1 /\ R2 and
A2: x in {_{X}_} by RELAT_1:def 13;
A3: [x,y] in R1 by A1,XBOOLE_0:def 4;
A4: [x,y] in R2 by A1,XBOOLE_0:def 4;
A5: y in R1.:{_{X}_} by A2,A3,RELAT_1:def 13;
  y in R2.:{_{X}_} by A2,A4,RELAT_1:def 13;
  hence thesis by A5,XBOOLE_0:def 4;
end;
