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