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
  f.:(X /\ f"Y) = (f.:X) /\ Y
proof
  thus f.:(X /\ f"Y)c=(f.:X) /\ Y by Th78;
  let y be object;
  assume
A1: y in (f.:X) /\ Y;
  then y in f.:X by XBOOLE_0:def 4;
  then consider x being object such that
A2: x in dom f and
A3: x in X and
A4: y = f.x by Def6;
  y in Y by A1,XBOOLE_0:def 4;
  then x in f"Y by A2,A4,Def7;
  then x in X /\ f"Y by A3,XBOOLE_0:def 4;
  hence thesis by A2,A4,Def6;
end;
