reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  (rng R) /\ Y c= (R~)"(R"Y)
proof
  let y be object;
  assume
A1: y in (rng R) /\ Y;
  then y in rng R by XBOOLE_0:def 4;
  then consider x being object such that
A2: [x,y] in R by XTUPLE_0:def 13;
A3: [y,x] in R~ by A2,Def5;
  y in Y by A1,XBOOLE_0:def 4;
  then x in R"Y by A2,Def12;
  hence thesis by A3,Def12;
end;
