reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  X /\ R"Y c= R"(R.:X /\ Y)
proof
  let x be object;
  assume
A1: x in X /\ R"Y;
  then x in R"Y by XBOOLE_0:def 4;
  then consider Rx being object such that
A2: [x,Rx] in R and
A3: Rx in Y by RELAT_1:def 14;
  x in X by A1,XBOOLE_0:def 4;
  then Rx in R.:X by A2,RELAT_1:def 13;
  then Rx in R.:X /\ Y by A3,XBOOLE_0:def 4;
  hence thesis by A2,RELAT_1:def 14;
end;
