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 Th82:
  rng(Y|`R) = rng R /\ Y
proof
  rng(Y|`R) c= Y & rng(Y|`R) c= rng R by Th78;
  hence rng(Y|`R) c= rng R /\ Y by XBOOLE_1:19;
  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;
  y in Y by A1,XBOOLE_0:def 4;
  then [x,y] in Y|`R by A2,Def10;
  hence thesis by XTUPLE_0:def 13;
end;
